Take the tour ×  The question is to find whether it converges or
diverges. If it converges, find what it converges to?  $$\sum_{n
= 1}^{\infty} \frac{2}{n^{1/3}} - \frac{2}{(n+1)^{1/3}}$$  I
tried splitting up into two summations but that didn't help.  I
think you can use the comparison test but if I use
$\large\frac{2}{n^{1/3}}$ to compare, it is a $p$-series with $p
< 1$, which diverges.  share|improve this question add comment
3 Answers  up vote 4 down vote accepted  HINT: Write out a few
terms and see that it telescopes.  share|improve this answer add
comment  You have already been given two nice ways to solve the
problem, so it is time for an ugly way.  Bring to a common
denominator. Multiply top and bottom by
$(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3}$ and use the identity
$x^3-y^3=(x-y)(x^2+xy+y^2)$ with $x=(n+1)^{1/3}$ and
$y=n^{1/3}$. Our expression simplifies to
$$\frac{2}{n^{1/3}(n+1)^{1/3}\left(
(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3}\right)}.$$ This is $\lt
\dfrac{1}{n^{4/3}}$.  Another way: The fol