Question

Check whether the sequence, {Sn} where

Sn=1/1!+1/3!+1/5!+....1/(2n+1)!
Is convergent or not.

Answer 1

$S=\frac{1}{1!}+\frac{1}{3!}+\frac{1}{5!}+...+\frac{1}{(2n+1)!}+...$

apply the d'Alembert ratio test to

$a_n=\frac{1}{(2n+1)!},\\ a_{n+1}=\frac{1}{(2n+3)!},\\$

consider

$lim_{n\mapsto \infty} \frac{a_{n+1}}{a_n}=lim_{n\mapsto \infty} \frac{(2n+1)!}{(2n+3)!}=\\ =lim_{n\mapsto \infty} \frac{(2n+1)!}{(2n+1)!(2n+2)(2n+3)}=\\ =lim_{n\mapsto \infty} \frac{1}{(2n+2)(2n+3)}=0<1.$

Since the limit value 0 is less than 1, the series is convergent.