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April 23rd, 2017, 08:47 AM   #1
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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
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April 23rd, 2017, 09:49 AM   #2
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$\dfrac{n^2+1}{n^3+n} = \dfrac{n^2+1}{n(n^2+1)}=\dfrac{1}{n}$

$\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ is the harmonic series which diverges.
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April 23rd, 2017, 09:53 AM   #3
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Quote:
Originally Posted by skeeter View Post
$\dfrac{n^2+1}{n^3+n} = \dfrac{n^2+1}{n(n^2+1)}=\dfrac{1}{n}$

$\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ 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 😊
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April 23rd, 2017, 09:57 AM   #4
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Quote:
Originally Posted by Lalitha183 View Post
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
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April 23rd, 2017, 06:52 PM   #5
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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!
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April 23rd, 2017, 07:45 PM   #6
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Try the integral test.
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April 23rd, 2017, 07:59 PM   #7
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Try the integral test.
Okay. Thank you
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April 29th, 2017, 01:09 AM   #8
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Quote:
Originally Posted by skeeter View Post
$\dfrac{n^2+1}{n^3+n} = \dfrac{n^2+1}{n(n^2+1)}=\dfrac{1}{n}$

$\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ 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.

Thanks in advance!
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April 29th, 2017, 05:57 AM   #9
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Quote:
Originally Posted by Lalitha183 View Post
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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