May 17th, 2017, 02:21 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 159 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: 2,604 Thanks: 1290 
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  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 159 Thanks: 1  Quote:
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  
Math Team Joined: Jul 2011 From: Texas Posts: 2,604 Thanks: 1290  Quote:
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: 2,604 Thanks: 1290 
I linked the wrong video ... meant to post this one. 
May 17th, 2017, 07:12 PM  #6 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 159 Thanks: 1  

Tags 
convergence, series 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Find the radius of convergence, and the interval of convergence, of the series.  Johanovegas  Calculus  2  January 23rd, 2016 01:34 PM 
Series convergence  klamtik  Calculus  1  May 28th, 2015 01:13 PM 
Convergence and Series Sum  Luiz  Real Analysis  4  May 10th, 2015 04:51 PM 
Series convergence  FreaKariDunk  Real Analysis  3  May 1st, 2012 07:13 PM 
convergence of series  Sambit  Real Analysis  10  January 4th, 2011 04:49 AM 