How to find limit?

Asked Dec 20 '16 at 23:49

Active Dec 22 '16 at 09:04

Viewed 73 times

Consider the sequence $\displaystyle a_{n} = \sum_{i = 1}^{n}i^{-1/3} $ and $b_{n} = n^{2/3}$.

No we want to find $\displaystyle \lim_{n\rightarrow \infty}a_{n}-\frac{3}{2}b_{n}$?

Actually I find it lower bound , but I used Abel summation, have no idea how to find it easier.

edited Dec 22 '16 at 09:04

Dominik

asked Dec 20 '16 at 23:49

openspace

Apply the integral test for bounds. – Simply Beautiful Art Dec 20 '16 at 23:50
@SimpleArt I want to know the easiest solution. – openspace Dec 20 '16 at 23:53
@SimpleArt I understand that I could use integral test, but I guess it's possible to simplify the sequence. – openspace Dec 20 '16 at 23:54
I'm betting it converges to $\zeta(1/3)$. – Simply Beautiful Art Dec 21 '16 at 00:00
@SimpleArt I get the same result , but that was a bit difficult. – openspace Dec 21 '16 at 00:02
I forgot where I saw it, but the limit of the sum minus the integral of this form converges to the zeta function – Simply Beautiful Art Dec 21 '16 at 00:03
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I think this has an answer: Limit related to $\zeta(x)$ – Sil Dec 21 '16 at 01:12

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