Hi everyone and welcome to Lillian's Vegan World. We're up to episode 13 and I am just loving the shows that we're doing. Today I have another awesome guest here to talk about the plant-based diet and the vegan lifestyle. The last episode I did was with Ellen who happens to be my guest today's other half. So I'm going to introduce you to Oahu's Captain Mushroom. Welcome, Vance. Thanks, glad to be here. Great, thank you so much for coming. I actually met your beautiful Ellen for the first time a couple of weeks ago and was very excited to hear that you are also not only vegan but producing some really excellent quality local produce here that we're definitely going to delve in and talk about. So welcome to the show. Thanks. You're quite fascinating actually because every time I meet a male vegan I kind of have to know how it all happened, where this journey started for you and every person I've talked to so far they actually all say the same thing that it starts from the woman that they're with. So I'm not sure if this is the case with you but we will find out. So Vance, you are Captain Mushroom, this is the name of your company that you have here and you're producing what? Oyster mushrooms primarily. Gourmet Oyster Mushrooms. That's great. If anyone saw the show last week you'll remember that we had some here in the studio and I was fortunate enough to take some of them home and sample them and I must say that they are excellent quality. I think everyone's going to be excited to get their hands on them but... What did you make by the way? So I did make stir fry with tofu and a plum barbecue sauce with walnuts and some cranberries that went over a bed of cauliflower rice. Oh. Those mushrooms I think this fantastic, very, very plain olive oil on the grill, salt and pepper, a bit of fresh lemon juice. You don't really need to do much to them to bring out the wonderful nesting. Yeah, I've definitely crucified a few batches myself by overcooking or an overspicing. Yeah, I don't think you need to. I noticed that Ellen, your partner, made the gyoza, the dumplings with those mushrooms. I would call that very high-end dumplings because you don't really need to use them in that way. So tell me a bit about yourself, Vance. Are you from here or where have you... How did you come to end up in beautiful Honolulu? That's a good story. Well, I was originally from Washington State. I spent the last six years down in the Virgin Islands, St. Thomas. Beautiful place. I had a charter boat company down there, Velocity Boat Reynolds. I had two boats, two people snorkeling, a lot of them had a snorkel and was living a good life. And then the two category five hurricanes hit us a couple of years ago. Yeah, I'm sorry to hear that. And really devastated the islands. Puerto Rico is still recovering. Right. And after several months of struggling with no internet hardly, I mean, we had Wi-Fi, but we didn't have Wi-Fi. It was not functional Wi-Fi. It, you know, just a lot of factors had changed. It wasn't as beautiful. There was a lot of chaos and problems. And the tourism was dramatically reduced and everything. So we were really struggling. My boats came through the Hurricane Ok. All the boats around me were devastated. And there was my boats after the hurricane and they were still floating. Gosh, that was lucky, although unfortunate for the other boats. But then our daughter moved over here to Hawaii and she was telling us how great it was. And so we decided to make the move and start something new. We didn't know what we were going to start. But you know, the beauty of capitalism is anybody can start anywhere and make, you know, something happen. If you want it bad enough, you can make it happen. Yes, that's awesome. It's really great to hear that even after this experience that you had to go through, you have come to a new island and to a new life and created something that probably wouldn't have happened had you not gone through that hardship. So yeah, great that you're having fun here and doing good things. But why did you choose mushrooms? It's very different to what you were doing. I wanted to get into something that Ellen and I could both be passionate about. And something that we could believe in and, you know, give back to the world in. And so, you know, I looked at hydroponics. I looked at vegan vending machine business. I was all ready to lay down, you know, for the vending machine business. Ellen didn't quite like that idea, you know, and so I kept looking and I was looking at hydroponics and then I came across mushrooms and I'm like, you know, and I always love mushrooms. I used to pick mushrooms in the wild in Washington State and the Cascade Mountains and up on Mount Rainier as well. And I just, you know, one thing led to another. I got an online course on how to grow them and started studying YouTube videos. And, you know, I'm like, okay, this is something that Ellen can get on board with and we can do and get back to the community and make a living as well. Awesome. Speaking to you just before we went on air, you were talking about how you grow them. It's a very complicated process and it takes a lot of really care and monitoring to get the mushrooms to grow to that, you know, quality that you are actually making them. So, let's just show our viewers now a couple of shots that you have sent in for us. The first photograph is, there you go. They are actually so huge. That one on the left there is eight and a half inches wide. Amazing. It's just kind of a monster. Elm oyster mushroom. Elm oyster mushroom. They do have a nice meaty sort of quality to them, don't they? Absolutely. Yeah, so they would actually be great for people to try as a steak or something like that. Oh, yeah. It's a nice vegan butter, garlic, a little bit of marinade, you know. That's how I like to enjoy them. But I've never seen them that vegan, hearty and meaty, so awesome. Big mushroom burger coming your way. Yes, yes, definitely. You know, if something like that was sold in a restaurant, so many people would be into that, I'm sure, but they don't really exist. Do they, food like that in restaurants? Oh, sure, you know. A nice huge, you know, oyster mushroom burger. I've never seen it before. Yeah. Over in Waimanello, I don't remember the name of the health food store over there. They have mushroom burgers right now, but they're Portobello's. Oh, okay, yeah. Yeah, Portobello's. They're delicious. Yes, they are. Let's have another look at your next one, Ben. It's their collage of terrible photos. You know, there's blue oyster mushrooms on the left there. Left middle is blue oyster mushrooms. They're called blues, but they start out blue and then they turn brown. And then just to the right of that, the very center is regular elm oyster mushrooms. The one in the lower right is actually a king oyster mushroom. Yes. It was a strange mix up with the lab. And these logs just wouldn't produce for weeks and weeks. Right. And I knew something was up. And I dropped the temperature. They finally sprouted and turned out to be the king oyster mushrooms, which they call them alihi's here in Hawaii. Alihi, yes, that's the Japanese name. Really? Yeah, it comes from the Japanese name. So the temperature is different for the oyster, the king oyster mushrooms as opposed to the elms, the temperature that they're grown in. What is the temperature for the elms, let's say? Around 65 is the optimum temperature for the oyster. For all your oyster mushroom varieties except for the king oysters, they like it down around 60 is even more optimum. There's some flexibility there, but there's optimum growing needs. Okay, excellent. You know what, Vance, we have to show your video because it will give the viewers a better idea of how you grow these amazing mushrooms. I think you're going to enjoy this. Take a look at Vance's video. Hey, are you mushroom lovers? Captain Mushroom here. Today, I want to show you my mushroom grow operation here in the heart of Waikiki. Right here is my incubation tent and over here is the fruiting tent. So first, we put the mushrooms in these logs. These are called logs. These are straw and coffee and you can see how the white mycelium is growing throughout here nicely. And then we poke these little holes in the bags and that's where the mushroom are eventually going to be coming out. These are blue mushrooms, blue oyster mushrooms and they'll be ready here in about another week to be moved into the growing tent. Now over here, we have the growing tent. The growing tent, we have our pond fogger that gives us our humidity and we have, it's very important to have the right humidity in here. Then I built this swamp cooler, homemade swamp cooler, blows cold air in here and keeps temperature at about 65 degrees. Now up here, I'm very excited to introduce my newest crop, the Italian oyster mushrooms. And you can see from a scale of my finger they've just popped out this morning and I'm very excited about them. And if you come in here real close, this is a very nice crop. I have to get in a little closer there. We've got the fogs kind of in the way a little bit, but yeah, these are, I'm very excited about these. This is my first crop of the Italian oysters and the last crop failed due to contamination so I'm very excited. These just came out this morning and I'm hoping to have them ready to ship out to the chefs on Friday or Saturday within the next two days. So to my Waikiki chefs, I'll be calling you soon. How awesome, that is such great footage and I myself as a chef have never seen anything like that before. Really? Yes, I learned a lot watching that. I've watched it several times just prepping for this show and it's very nice footage, yeah. I have to mushroom on a mission. So what did you think of the title? Vegan superhero taking mushrooms to a whole new level in Oahu. Well, that's very flattering. But I like what you said at the end. Waikiki to the Waikiki chefs, I think I make a living cooking as well and these are going to fly out of your tent once you start getting really good production going. Vance, where can we get a hold of some of these mushrooms? How can we get a hold of them? You just pretty much got to call me. My wife and I, we're a one show operation here. There's our phone number,<PHONE_NUMBER>. Awesome. And you're located now in, is it Kailua? You're over in Kailua? We're on the west side, yeah. On the west side, yeah. Okay, great. So get in touch. Get in touch and get some gorgeous mushrooms from Captain Mushroom. They come highly recommended. Let's have a look at another one of Vance's photos. Which one is this, Vance? These are the golden oyster mushrooms. Yes, they're all very different in color, aren't they? A bright golden mushroom. This particular photo didn't turn out very clear and for some reason we had lost the other photos that we took. They're a little different. They say they taste a little bit like chestnut. Yes, I was going to say I find them to be a little bit more earthy. Okay, you've had them. Yes, in flavor, yes. In Japan, not never here. I've never seen them here, I don't think. In Japan, they carry a very large variety of mushrooms and eat a lot, consume a lot of mushrooms. So that's where my love for these fungi or fungus started. But I love mushrooms too. When we get back from the break, we're going to talk a little bit about the nutritional value and how mushrooms are good for you. But basically they're tasty, so that's why we should be eating them as well. What's your favorite mushroom dish? I like, Helen makes a mushroom soup to die for. Mushroom soup, I love that too. There's something so comforting about a big bowl of chunky, creamy or whatever mushroom soup. Oh yeah, the mushroom is crunching on your palate and absorbing into your... Yes. Stay tuned, we've got lots more to talk about with Captain Mushroom. I'm just going for a quick break and see you after it. Hello. Thank-Tek Hawaii is live in Japan. Hello, I'm White, host of Kunisei Ikari. Every Monday, Monday to Monday, we have a host, in the audience, in the Japanese language. Hi, I'm Rusty Kumori, host of Beyond The Lines. I was the head coach for the Punahou Boys varsity tennis team for 22 years, and we were fortunate to win 22 consecutive state championship. This show is based on my book, which is also titled Beyond the Lines, and it's about leadership, creating a superior culture of excellence, achieving and sustaining success, and finding greatness. If you're a student, parent, sports or business person, and want to improve your life, and the lives of people around you, tune in and join me on Mondays at 11 a.m. as we go Beyond the Lines on Think Kakaway. Aloha. Welcome back, everyone, to Lillian's Vegan World, the show where we talk about everything veganism and the plant-based lifestyle. Today, I have my awesome guest, Captain Mushroom, who is taking Mushrooms to a whole new level right here in Oahu. We just showed some of Vance's beautiful photos and footage from the place where he makes all this magic happen and creates these beautiful mushrooms right here in Honolulu. So do contact Captain Mushroom if you do want to get your hands on some of these gorgeous things. Vance, I would like to talk a little bit about what I have been doing on the island. So if I could have my first photo. This month is all about Serbian cuisine. In all of my Vance, I'm doing vegan Serbian cuisine. This is some vegan feta that I made using tofu, which resembles the texture of regular feta cheese. So it's a good way to make the vegan version. My dad was actually Serbian and that's why I have this love affair with Serbian food. So I use feta a lot, the vegan feta, and this is one of the recipes that I was doing in my cooking demo at Down to Earth yesterday. Have a look at another one of my pictures. As you all know by now, I'm a vegan chef and cooking instructor. I also do private catering for families and this is every week I do cater for a couple that asked me to prepare five dishes once a week. This is an example of one of the things I did. The bottom left Vance has your king oyster mushrooms in it. This is the one with the plum sauce that has a bit of a barbecue sauce concoction going on. Some tofu and kale, lots of good stuff there. Roasted, what have we got at the back there? What are those vegetables called at the back of Vance? Take a guess. I can't think of the word in English. Brussels sprouts. In Japanese the word is mere cabbage. That means small baby cabbage. And that's the word that was coming. I couldn't think of the word in English. But anyway, there's an idea of... And the soup actually in the middle is a mushroom soup, a creamy mushroom soup with corn that I made. Let's have a look at one more. This is my Serbian menu for this month. Top left is a biver, which is a red bell pepper sauce. That is very, very popular over there in Serbia. Some beans, also beans, very popular there. Spinach and tofu, phyllo pastry pie. Again, everything has feta cheese in it. Down the bottom right, which you and your beautiful wife Ellen are going to sample, is some Serbian cornbread with broccoli and vegan feta cheese. So very interesting flavours going on in Serbia. Would you say it's spicy food? No, not at all. It's actually quite mild on the palate. Very similar to all the countries in the Balkans in that area all have similar dishes using lots of paprika or paprika. Yeah, lots of feta cheese again. But Serbia is definitely a meat-eating country. So had my dad still been here with us today, I think he would have been... I don't know if the word is... I don't know what the word is, but probably very surprised I'd gone and veganised his whole country's cuisine. But you've got to do what you've got to do. For the planet. For the planet, yes. And for our health. And for our health, yes. So one more of mine, if you don't mind, Haley. This is some upcoming events. So next week I have more cooking demos featuring the vegan Serbian cuisine. My seven course dinners as usual. And next month I'm going to be doing four course vegan brunch as my cooking demos and cooking classes. And also we'll be doing that down to earth. So lots of vegan events I'll be holding and looking forward to. And thank you to everyone down to earth who came yesterday to my demos. I really appreciate it. So captain mushroom, getting back to mushrooms. They are actually very good for you. Oh, absolutely. Yes, very good for you. So I did prepare a little bit of a nutritional profile of the mushrooms. Most mushrooms, there's obviously a large variety, but basically they all have about the same sort of nutritional value and calories. So about one cup of chopped or sliced mushrooms comes to only 15 calories. Zero grams of fat. 2.2 grams of protein. 2.3 grams of carbohydrates. And it also includes 0.7 grams of fiber and 1.4 grams of sugar. So basically there are five things that I found according to most research and studies. Basically mushrooms are known to lower cholesterol levels, alleviate inflammation. They are also obviously packed with antioxidants. May block cancer growth and boost brain health. Yeah, and your immune system. Oh yeah, the lion's mane mushroom in particular. They're used for all-timer patients and as well as dementia patients. They're getting positive results. Yes, definitely. So this boosting brain health that I think is something that maybe not a lot of people know about. Maybe people should be eating more mushrooms or incorporating more mushrooms into their diet just to be sure. Oh yeah, we had some lion's mane, Ellen and I while back and we were just sharper. We could remember things quicker. Oh really? We noticed a significant difference. Okay, awesome. I might have to up the ante on the mushrooms. There was an article by the Medical News recently that talks about mushrooms and as regards to cancer, what you just mentioned earlier on mushrooms are very high in antioxidants, just like carrots, tomatoes, green and red peppers, pumpkins, green beans, zucchini and other whole foods. Antioxidants are chemicals that get rid of free radicals, a type of chemical that can harm a person's body cells potentially leading to cancer. Did you notice fans that in this little quote that I just mentioned, nothing says that you need to eat more animal product? Nothing there for that. They don't need it. Our bodies weren't, the creator didn't design our intestines with, you know, as much length that we have to be eating meat, you know. Yeah, that's right. The shape of the teeth, there's no fangs. The claws, not strong enough to really, a baby would not be able to attack a chicken on its own and get into it probably, but would eat an apple that had fallen off the tree. Yeah, all the carnivore animals and testines are way shorter than ours. Yep. You did mention to me earlier on fans that, let me go back a step. How did you become vegan? Well, Ellen got me into veganism. She started cooking. Waved the magic wand with her little wooden spoon. That's how we do it fans. We cook for you guys and then you fall in love. Did you not know that? Well, it worked anyway and you ended up with your beautiful wife. So she sort of, she was vegan before you became vegan. She was, she eased me into it. She's like, you know, I'm like, I could really use a steak. I haven't had a steak since we've met, you know, because she was cooking almost every night.  And so she was, she's like, we're non-judgmental, you know, and she cooked me the steak, even though she's like, it fills me inside, but I'll do it, you know. That's love. Yeah. And so, I mean, she eased me into it.  So, you know, all this good vegan food. And I was just so surprised that vegan food could be so delicious. Right. You know, I always thought it was rabbet food, you know. And yes, yes. I think, I think that stereotype is slowly starting to disappear now that more and more vegan food is becoming, you know, sort of mainstream. And a lot of people, a lot of fast food restaurants. Have you seen the Burger King just came out this week? Yes, with the impossible burger. This week, we had him, we went straight down and drove the first time we've been in the drive-up window in a few years now. Uh-huh. Yes, I know. How was it? Oh, they were just like a whopper. Delicious. Flame broiled taste. You couldn't tell the difference. In fact, Helen's like, I think they put meat in here. They're selling them as possible because they ran out of them. Yeah. So a lot of people are saying that. Because they taste the same. I know yesterday I was at Down to Earth. I had just finished my demo. Sitting outside, I was drinking iced tea. A person next to me had bought the impossible burger and was sitting down eating it and just going through how, what he thought about it. And he said exactly the same thing. He's like, dude, I swear there's meat in this. I hope they didn't make a mistake. Yeah. Well, convincing this product is, I mean, it must be very, very close. What's the other mean? Beyond. The Beyond burger also is exceptionally delicious. Yes. And that's so great. I mean, there are some people, there are some vegans who sort of criticize that sort of food, calling it unhealthy or a bad example of vegan food. But in my opinion, I support these products, whether I eat them or not. They're awesome for people who are transitioning or people who are trying to learn how to eat more plant-based food. So I definitely support all of that alternative food. When you're at a barbecue, you know, it's good to just throw a couple vegan patties on the barbecue and join the party. Absolutely, yeah. And that, yeah, you feel more part of the social team, don't you? Fans, see how quickly the time's gone. We are running out of time and do have to close. Would you like to say something quickly before we close? Well, I don't know. I mean, we're starting out small right now. But we'll be having more mushrooms available in time. So if you want to give us a call, our phone number is<PHONE_NUMBER>. And give us a call. And will we either have mushrooms in or we don't. We're applying a couple of restaurants already. So we may have them. But within the next few months, we'll have a lot more mushrooms. We'll be a lot better off. And you'll be lucky to get your hands on them before I do, because I'll be there calling you for more of your excellent produce. Thank you, everyone, for watching Lillian's Vegan World. I hope to see you in a couple of weeks. Have a great weekend and aloha.


I'm Tom Handel here with another CTN member highlight and this time we're featuring Independence Association. They're doing a fundraiser called A Run for Independence coming up soon and I have here with me Dennis St. Pierre, the director of marketing and community relations, right? Right. Before we get into talking about the Run for Independence, could you tell us a little bit about what Independence Association is? Independence Association basically is a servicing agency for people with disabilities. They could be developmental, mental, intellectual, so forth. But we're directly engaged in trying to create efforts towards inclusivity for all members so that they truly are part of society as a whole and part of their community. So our organization works with different organizations. We've created new organizations under our umbrella, all with the intent of creating inclusivity for our members. And how far do you reach and how many clients do you serve? Well, we reach throughout the whole mid-coast area and the Brunswick area, mostly. Brunswick, Topsom, we're into Durham, we're in Bath, and a few other locations, Freeport. And we have, we probably service, I would say, anywhere between 200 to 400 people. It may be even more than that. I'm new in this, so I'm still learning. What is it makes sure of residential services and day versus employment services? We have day programs, residential, we do all kinds of different community efforts. We have Spindleworks, which most people know about. That is one of our agencies. And we've just tied off into another one in the gardener called Spinoff. So that's a arts type of group where our individuals can come in and do art and work with artists, professional artists of all different levels and different types. That's great. And it's a really essential kind of a service that people need in these areas. Now, you get funding from the state, but I understand there's often problems with being fully funded for all the things that you do. Right. And especially now with the federal budget cuts and everything else, we're looking at what is going to be happening in the future. And we know that we're going to have to start raising funds and awareness of what we do and the services that we offer to our community. Because it's not just to our members that we're servicing. We're servicing our community as well. And we want to make sure that they know we're there. So if they know we're there, then we can actually ask them to help support us financially. And we know that's coming, that we're going to need to do that more. So fundraising is really important for you. And you've got this run for independence coming up now. It's not the first time you've done this, right? No, last year it was called the Sea Dog Dash. And that was our first annual run. We had 485 people run in the race for first race. It was fantastic. It's a 10K, a 5K in a one mile. And this year we're adding a kids run as well. So it starts, it's on May 18th. It starts at 9am in Brunswick at the Brunswick Junior High School. And this year we're expecting anywhere between 500 to 700 runners. Wow. That's what we're hoping. And the effort's going pretty well. So now people you're looking for people to run in the race. You're looking for sponsors. Right. What's involved in doing all of that if somebody wants to be a sponsor or run in the race? If they want to be a sponsor, all they need to do is contact me through the Independence Association. Dennis, well, it's actually D Saint Pierre, one word at iaofmain.org. And you can contact me through that for sponsorship. And if you want to register to actually race, you can go into the website called run in a race.com. And you can register right there online. And how do people, how do you actually raise money when people run? And what's the for people who aren't familiar with that kind of fundraiser? Well, we basically raise the money through fees and through sponsorships. That's how we do the fundraiser. The fee to race and the race is very reasonable. If you sign up before May 1st, it's a discounted rate. And as we get close to the race, it goes up. But it's extremely reasonable in price for people to register to race. And you can see that on the registration form, the prices are listed there. And if people want to make donations, they can do that as well. And we've also got a great opportunity with Andrew Scogging Bank, who is offered to be one of our sponsors. And they're doing a promo with us. Anybody who wants to sign up for their mobile app, and, and says that Independence Association, the run for independence referred them. Andrew Scogging Bank will make an extra donation of $25 to the run. And that's for anybody who uses says run for independence or Independence Association in relation to signing up for that mobile app. And the mobile app is free for any person who has an Andrew Scogging Bank savings account or checking account or anything. They we would get that. No, is there a deadline for sponsors to sign up or runners to sign up? We're really under really tight schedule. As you can see, May 18th is only three weeks away. If certain sponsorships, we are already at the deadline today. If you want to have your logo on things and so forth like that and the promotional materials. But we're accepting sponsorships right up until the 18th of May. So if people want to sponsor and donate, we would love that. And we will make sure that the name gets put on the website, and the website will stay up until next year. So whoever sponsored those names will be up there in perpetuity until we start working on next year's race. Great. Well, just real real quick. Once again, where do people go if they want more information about how to become a sponsor or run in the race? For Run for Independence, you can go to run for independence.org. We have our own website and there's more information there in links to our our actual nonprofit Independence Association and to run in a race.com. Great. Great. Well, thanks for joining us. Thank you very much, Tom.


Let us determine the convergence of the series where we add together from n equals one to infinity sine of n over factorial. Hmm, this one's kind of a tricky one and I want you to, I want you watching this at home to kind of pause the video for a moment and actually think about what would I do to determine the convergence of this one? And I want you to be disciplined that I don't want you to unpause this video and finish watching until after you have a good idea on what to do. Hey, I told you not to unpause the video until you figured this one out on your own here. At least you gave it a good, honest try here. So you pause that video and do it for real. All right, so assuming that we all took a good stab at this one, we might have, some of us might have been successful, some of us might have got stuck because like I said, this one is a tricky one, right? So what's going on here? This, really the obstruction is coming from the sine of n on the top because unlike many series, this is not a positive series, right? Sine of n, right, this is sometimes positive but it's also sometimes negative. Potentially for different choices of sine, this thing will be jumping between numbers, we'll be choosing real numbers between negative one and one. But I'll never, so it's not a positive sequence. It's not a positive sequence. And so that's an important thing to make mention here because the fact that it's not positive means that we can't use the integral test, even if we could find the anti-derivative sine of n, even if you could find the anti-derivative sine of x over x factorial, whatever x factorial even means, but the integral test does require that this thing become eventually a positive function. It also has to be eventually continuous and eventually decreasing, but sine of n will never eventually be positive because it will be equally likely negative as it is positive. So the integral test doesn't apply here. Neither does the comparison test. The comparison test, and I'm including the limit comparison test here as well. The limit comparison test only works when we compare a positive sequence to a positive sequence. And then when there's an appropriate comparison there, we can then say something about the convergence of the series. So the comparison test and the integral test does not apply here. Now oftentimes we could use the alternating series test but this series is not an alternating series. It's not alternating series, right? Which that tells us that the alternating series test doesn't apply here as well. Now an alternating series does one of two things. It either goes positive, negative, positive, negative, positive, negative, positive, negative, or maybe it starts negative, positive, negative, positive, negative, positive. You have this back and forth every other time. That's not what this series will do. This series will do something like positive, positive, negative, positive, positive, positive, negative, negative, positive, negative, positive, or something like that, right? It's not an absolute every other time. So a lot of the convergence tests that we know about don't apply. It's clearly not a P test because of the sign of N on the top. It's not a geometric series so that doesn't apply. So we're actually showing a lot of options that we can't do. We cannot use these things, all right? So again, this is getting super depressing here. What is left? What could we still use? Well, we potentially have the ratio test, don't we? The ratio test, it doesn't have to be a positive or alternating series. So potentially have the ratio test. But if we were to use the ratio test, what are we gonna have? We're gonna be looking at the sequence A N plus one over A N. And so that would look something like the following. Sign of N plus one over sign of N times N factorial over N plus one factorial. Now the factorials aren't a problem for the ratio test. This would just simplify to be one over N plus one. But this creature right here, sign of N plus one over sign of N. If we wanted to determine whether that thing was, what's the limit there, right? What's the limit as N goes to infinity of sign of N plus one over sign of N. The problem is sign of infinity is not defined. There's no limit there because of the oscillating nature of the function. So one has some concerns about what's gonna happen here, right? What is gonna happen to this thing? Now, this limit calculation is not gonna be super helpful for us if we try to resolve it, right? Now again, we actually have to calculate in with the idea there's an N plus one on the bottom. That does actually give me a little bit of hope that this thing right here, right? Sign of N plus one over N plus one. Maybe that'll give us a home up, oh wait, that's if N goes to zero, right? That equals one. If N goes to infinity, feels like something's going on here. The issue is one could make the argument that since sign is bounded above, it's always between negative one and one like we saw before. The denominator is gonna get bigger and bigger and bigger in terms of absolute value, although there's some oscillation going on here. This thing, we could try to make an argument that this thing is gonna go off towards zero maybe, right? One could do that, but that's gonna be kind of a difficult limit to do for a calculus student, possible, but we could perhaps make that work. We could also try the root test. So I guess what I'm saying is the ratio test might have some promise, but it's gonna be a difficult limit. One could try the root test instead, but you're gonna see that in my opinion, the root test might actually be somewhat worse in this situation, right? Because we have to take the nth root of the absolute value of sign N, N factorial. That in my opinion looks like it's gonna be even worse. How do you deal with the nth root of sign? So no, the ratio test, if you're gonna go this direction, the ratio test is something you could do. But there's another, I think, much easier thing to show here that some people often forget about, okay? So one thing to remember is that if a series is absolutely convergent, then it's convergent, right? Absolute convergence, this implies convergence. And why this is worth considering is the following idea. If we take the absolute series, so we take the sum where n equals one to infinity, we're gonna get the absolute value of sign of N over N factorial. Now N factorial is always positive. So in terms of the absolute value, that's only the absolute value of sign, right? And so when we take this absolute value sequence, this absolute sequence, one thing I wanna mention now is that this is now a positive sequence. So some of the tools that we had to throw out earlier actually applicable to the absolute series. For example, the integral test applies now. I don't wanna do the integral test because again, this would be a horrible anti-derivative. But maybe a comparison test would actually work, right? That's kind of important to know. The absolute value of sign of N, this will be bounded below by zero, since sign, of course, since we're taking average value as positive now, but it's also bounded above by one. And so we could then compare this to the series one over N factorial where N equals one to infinity, which this is a series that we have looked at before. This is one that you're gonna wanna know. I mean, this series right here, well, you could use the ratio test very simply to do this series is what I guess I'm trying to say here. For this one, your limit here, you're looking at this N plus one, I guess it's actually, the limit you're looking at here would be N factorial over N plus one factorial, which that simplifies to be one over N plus one, which that converges towards zero here, which is less than one. So the ratio test would apply convergence here. Ratio test implies convergence here. And then the comparison test implies convergence here. And since this is the absolute series, the convergence of the absolute series implies the convergence of the original series. And so this shows that the original series is convergent. And if I had a quote, a test here, we'll call this by the test, the test of absolute convergence because absolute convergence is an easier thing. And one reason we like absolute convergence is when something is positive, the series is entirely positive, we have more options to the test, it's convergence or divergence than if something is sometimes positive, sometimes negative. So there is an advantage by looking at the absolute series in this case. And that's actually the approach I would recommend. You could have started off with the ratio test without using the absolute series. And then actually give us the absolute convergence of the series, which is great. That is good to know here. So kind of compare these different techniques here. Again, for your typical student absolute convergence is the direction to go here. I think that'll be much more helpful. But when you're approached with a series and no one's telling you which convergence test to use, remember, this is a street fight. We can use whatever tools, whatever weapons we have available for the fight. And so the only rule is that if you can do it, then you can do it. And so try to think through these problems, try to try different approaches. Some will be successful, some won't. You could try to analyze it like we did in this video. Why can't I use this tool? Why could I use this one? And as you try to argue with yourself with these, you'll really find a deeper understanding of the series and why they converge and why they sometimes diverge.