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 April 23rd, 2017, 07:47 AM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 153 Thanks: 1 Convergence Series Can anyone let me know Is my assumption of the following series is convergent is true ? $\displaystyle \sum_{i=1}^{\infty} \frac{n^2 + 1}{n^3 + n}$ Many thanks in advance
 April 23rd, 2017, 08:49 AM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 2,575 Thanks: 1274 $\dfrac{n^2+1}{n^3+n} = \dfrac{n^2+1}{n(n^2+1)}=\dfrac{1}{n}$ $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ is the harmonic series which diverges.
April 23rd, 2017, 08:53 AM   #3
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Quote:
 Originally Posted by skeeter $\dfrac{n^2+1}{n^3+n} = \dfrac{n^2+1}{n(n^2+1)}=\dfrac{1}{n}$ $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ is the harmonic series which diverges.
Thank you so much for the reply. Can you let me know how to check whether a sequence is convergent or divergent.

Thank you 😊

April 23rd, 2017, 08:57 AM   #4
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 Originally Posted by Lalitha183 Thank you so much for the reply. Can you let me know how to check whether a sequence is convergent or divergent.
There are quite a few tests for convergence ...

list of series tests

April 23rd, 2017, 05:52 PM   #5
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Quote:
 Originally Posted by skeeter There are quite a few tests for convergence ... list of series tests
Thanks for the link. I have one more doubt! Can we use any one of the provided tests to check the series convergent or divergent?

Thanks again!

 April 23rd, 2017, 06:45 PM #6 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,514 Thanks: 909 Math Focus: Elementary mathematics and beyond Try the integral test.
April 23rd, 2017, 06:59 PM   #7
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 Originally Posted by greg1313 Try the integral test.
Okay. Thank you

April 29th, 2017, 12:09 AM   #8
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Quote:
 Originally Posted by skeeter $\dfrac{n^2+1}{n^3+n} = \dfrac{n^2+1}{n(n^2+1)}=\dfrac{1}{n}$ $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ is the harmonic series which diverges.
I have a doubt. How Infinity works in Series ?

Can you help me in understanding its properties like what could be the value of $\infty / \infty$, $\infty * \infty$ or $\infty+ n$ ($n$ is some natural number).

Please clarify its values to determine the limit.

April 29th, 2017, 04:57 AM   #9
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Quote:
 Originally Posted by Lalitha183 I have a doubt. How Infinity works in Series ? Can you help me in understanding its properties like what could be the value of
$\infty / \infty$ is indeterminate

$\infty * \infty = \infty$

$\infty+ n = \infty$ ($n$ is some natural number).

Go to the link for further information ...

The Algebra of Infinity

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