POTW Proof of Asymptotic Equality: $\sum_{n=0}^\infty a_n\frac{x^n}{n!} \sim ae^x$

  • Thread starter Thread starter Euge
  • Start date Start date
Euge
Gold Member
MHB
POTW Director
Messages
2,072
Reaction score
245
If ##(a_n)## is a sequence of real numbers with ##\lim a_n = a##, show that $$\sum_{n = 0}^\infty a_n\frac{x^n}{n!} \sim ae^x$$ as ##x\to \infty##.
 
  • Like
Likes topsquark and malawi_glenn
Physics news on Phys.org
For any ##\epsilon >0## there exist ##N## such that if ##n>N##, ##|a_n-a| < \epsilon##.

|(RHS-LHS)/RHS |=
|\frac{\sum_{n=0}^\infty (a-a_n)\frac{x^n}{n!}}{ae^x}|&lt; \frac{\sum_{n=0}^N |(a-a_n)\frac{x^n} {n!}|+\epsilon \sum_{n=N+1}^\infty \frac{x^n}{n!}}{|a|e^x}
= \frac{\sum_{n=0}^N (|a-a_n|-\epsilon)\frac{x^n} {n!}+\epsilon e^x}{|a|e^x}\rightarrow \frac{\epsilon}{|a|}
##\epsilon## can be taken as small as we like. So the given asymptotic equality is proved.
 
Last edited:
  • Like
Likes malawi_glenn, Greg Bernhardt and topsquark
Back
Top