Determine whether the series ∑_(n=1)^(infinity) ((squareroot(n)+ln(n))/((n^2)+1)) converges or diverges. clearly identigy any test(s) you are using and label all relevant data and/or properties.

I have no idead where to start or what tests to use.

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- Oct 24th 2012, 01:45 PMPreston019Help With Series
Determine whether the series ∑_(n=1)^(infinity) ((squareroot(n)+ln(n))/((n^2)+1)) converges or diverges. clearly identigy any test(s) you are using and label all relevant data and/or properties.

I have no idead where to start or what tests to use. - Oct 24th 2012, 02:18 PMbenb89Re: Help With Series
It can be shown that $\displaystyle \ln(n) \leq \sqrt{n}$

Can you then find a bound on each term of your series?

Hint - think comparison test and p-test to arrive at your final answer! - Oct 24th 2012, 02:46 PMPreston019Re: Help With Series
- Oct 24th 2012, 03:01 PMbenb89Re: Help With Series
So,

$\displaystyle \frac{\ln n + \sqrt{n}}{n^2 + 1}\leq \frac{\sqrt{n} + \sqrt{n}}{n^2 + 1} $

Can you go forward from here? - Oct 24th 2012, 03:04 PMPlatoRe: Help With Series
NO!

$\displaystyle \frac{\sqrt{n}+\ln(n)}{n^2+1}\le\frac{2\sqrt{n}}{n ^2+1}<\frac{2}{n^{3/2}}$.

The last is a p-series that converges. - Oct 24th 2012, 03:05 PMPreston019Re: Help With Series
- Oct 24th 2012, 03:08 PMPreston019Re: Help With Series
- Oct 24th 2012, 03:21 PMPlatoRe: Help With Series
- Oct 24th 2012, 03:33 PMPreston019Re: Help With Series