Answer to Question #109844 in Real Analysis for Pappu Kumar Gupta

Question

Check whether the sequence, {Sn } WHERE
,Sn =1/1!+1/3!+1/5!+...+1/(2n-1)!


Is convergent or not.

Answer 1

Use the Ratio Test

$$\lim\limits_{n\to\infin}\big|{a _{n+1}\over a_n}\big|=\lim\limits_{n\to\infin}\bigg|{\dfrac{1}{(2(n+1)-1)!}\over \dfrac{1}{(2n-1)!}}\bigg|=$$

$$=\lim\limits_{n\to\infin}\big|{1\over 2n(2n+1)}\big|=0$$

The sequence is convergent by the Ratio Test.