In this video we provide the solution to question number 11 for practice exam number 4 for Math 1220. We're given the series where we take the sum where k equals 2 to infinity of the sequence 1 over k times the natural log of k squared. And there's two parts to this problem here. So the first part is that the series can be, so just reading the problem here, the series can be found convergent using the integral test. And so we're going to verify this by computing the integral from n to infinity of 1 over x times the natural log squared of x dx. So notice that to show that this thing was convergent, we'd show that this integral is convergent specifically when you have a 2 right here. But we're going to do this in more generality, we're going to do it with an n. So that's where we want to compute this integral. Okay, so the integral from n to infinity here, 1 over x times the natural log squared of x dx. A u substitution is very relevant here. You could take u to be the natural log of x. So then du would equal dx over x for which then our integral would then become, you're going to get the 1 over x dx is going to become your du. And then the denominator, you have 2 natural logs, so you get a u squared. And so which case this is in the natural log of u. Excuse me, when you plug in n inside here, you're going to get natural log of n. So u is the natural log of n on the bottom and then at the top it's still infinity. As x approaches infinity, u will approach infinity as well. So using the power rule here, the anti-derivative is going to be negative 1 over u as you integrate from natural log of n to infinity here. Because of the negative sign, I'm going to switch the bounds around. This is going to be 1 over u as you go from the natural log of n on top to infinity on the bottom. And so then plugging that in here, you're going to get 1 over the natural log of n minus 1 over the natural log of infinity. For which the natural log of infinity is itself infinite. So you get 1 over the natural log of n minus 1 over infinity. That last term is then 0. And then simplifying, you end up with 1 over the natural log of n. And as n is any positive integer, this is going to be a positive number. And since this is not infinity, that would then show that this integral is a improper or this improper integral always converges. And so that's how you'd use the integral test. So because this integral is convergent, this series is convergent as well. Now we integrated this from n to infinity as opposed to 2 because we're not just after the convergence. We're also after the error bound. So this right here gives us an error bound with regard to the integral test here as we can use this to approximate things. So this gets us to the second part of the problem. What is the smallest choice of n that would guarantee that the partial sum where you take the sum where k equals n to k equals 2 to n of that same sequence 1 over k times the natural log of k squared there would be how what n guarantees that the partial sum approximation is accurate to within 1 over the natural log 100. For which you can use a calculator to approximate that, but I chose a number here specifically so you can do this without a calculator. So even if you forgot your scientific calculator, we can do this thing. So using the error bound with regard to the integral test, the error of truncating after the nth term is bounded by honestly the term we just found. It's going to be bounded above by this, which is that integral. So the error we want is the following. We take the error bound we found, which is 1 over the natural log of n. This needs to be less than or equal to 1 over the natural log of 100, like so. Taking the reciprocals, we need that the natural log of n. We need that to be greater than or equal to the natural log of 100. Taking the exponential of both sides, we need that n needs to be greater than or equal to 100. And so if we take the first 100 terms of the sum and add those together, that will then be accurate within this error that's described here.


What's going on guys it's your boy DPSH here today with another destiny video and today wow wow just fucking wow now before we go any further I gotta say at the moment this has not been confirmed but these pictures do look as genuine as any of the others that have been leaked over the past week regarding the rise of iron take a look at this this image has surfaced well these images have surfaced over the past few hours and it clearly shows a new version of the also beloved and most famous gun in any game ever a new version of the gala horn it's called the iron gala horn lo and behold if you want this beast it comes with the pre-order of the rise of iron expansion absolutely insane again it's called the iron gala horn and it looks beautiful now I've seen two separate images confirming this and at the moment it is believed them to be legit but still take it with a pinch of salt because like I said this is not being confirmed by Brunjie the small print and the image on the left reads access to the iron gala horn starting September 20th 2016 can be attained by pre-ordering destiny rise of iron and completing the in-game quest CWWW destined to the game that come for additional details the iron gala horn not exclusive to any retailer or platform subject to availability so this states the weapons obtained by pre-ordering rise of iron and completing a special quest sounds rather interesting could that quest take us to the battle of the twilight gap maybe not but wishful thinking don't hurt anybody but yes guys the iron gala horn it seems to be legit if it is well what can you say the gal was making a return absolutely epic but still until this is confirmed don't get too hyped take it with that pinch of salt but if it is legit this is how you get it pre-order the rise of iron and completing that special in-game quest and you will be rewarded the iron gala horn absolutely epic but guys hoping to do the video do drop a like I do appreciate the support tell me your thoughts on this weapon down below in that comment section and I'll catch you on that next one peace out till next time