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 May 17th, 2017, 02:21 AM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 106 Thanks: 1 Series Convergence The set of all values of $a$ for which the series $\displaystyle \sum_{n=1}^{\infty} \frac {a^n }{n! }$ converges is A) $\displaystyle (0,\infty)$ B) $\displaystyle (-\infty,0]$ C) $\displaystyle (-\infty,\infty)$ D) $\displaystyle (-1,1)$ I have two opinions for this question. Option D - As it is convergent at $0$ and also around $0$ for the given $n$ values. Option B- As I know for $a= -1$ it is convergent but for the other values of $a$ like $-2,-3...$ does it converge ? Please clarify and also rectify if I'm wrong. Thank you
 May 17th, 2017, 03:04 AM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 2,508 Thanks: 1234 using the ratio test for convergence ... $\displaystyle \lim_{n \to \infty} \left| \dfrac{a^{n+1}}{(n+1)!} \cdot \dfrac{n!}{a^n} \right| < 1$ $\displaystyle |a| \lim_{n \to \infty} \dfrac{1}{n+1} <1$ $|a| \cdot 0 < 1$ for all values of $a \implies$ interval of convergence is $(-\infty,\infty)$
May 17th, 2017, 03:35 AM   #3
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 Originally Posted by skeeter using the ratio test for convergence ... $\displaystyle \lim_{n \to \infty} \left| \dfrac{a^{n+1}}{(n+1)!} \cdot \dfrac{n!}{a^n} \right| < 1$ $\displaystyle |a| \lim_{n \to \infty} \dfrac{1}{n+1} <1$ $|a| \cdot 0 < 1$ for all values of $a \implies$ interval of convergence is $(-\infty,\infty)$
I think rario test is used for finding the convergence of the series not for the values of $a$ ?
I'm confused. Can you please explain where the series is convergent in which intervel ? and at which values of $a$ ?
Thanks

May 17th, 2017, 04:01 AM   #4
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Quote:
 Originally Posted by Lalitha183 I think rario test is used for finding the convergence of the series not for the values of $a$ ? I'm confused. Can you please explain where the series is convergent in which intervel ? and at which values of $a$ ? Thanks
The given series converges for all values of $a \in (-\infty,\infty)$. The correct choice is C.

Watch the linked video to see another example of using the ratio test to determine an interval of convergence.

 May 17th, 2017, 05:10 AM #5 Math Team   Joined: Jul 2011 From: Texas Posts: 2,508 Thanks: 1234 I linked the wrong video ... meant to post this one.
May 17th, 2017, 07:12 PM   #6
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Quote:
 Originally Posted by skeeter I linked the wrong video ... meant to post this one.
I understood now. Thank you

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