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May 17th, 2017, 02:21 AM   #1
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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.
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May 17th, 2017, 03:04 AM   #2
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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 $(-\infty,\infty)$
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May 17th, 2017, 03:35 AM   #3
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Quote:
Originally Posted by skeeter View Post
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 $a$ ?
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May 17th, 2017, 04:01 AM   #4
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Quote:
Originally Posted by Lalitha183 View Post
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.

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May 17th, 2017, 05:10 AM   #5
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I linked the wrong video ... meant to post this one.

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May 17th, 2017, 07:12 PM   #6
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Originally Posted by skeeter View Post
I linked the wrong video ... meant to post this one.

I understood now. Thank you
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