Answer to Question #161758 in Calculus for Vishal

Question

Prove that if ∞n=0an is a convergent series, then limn→∞an = 0 (i.e., starting with n = 0 in the series yield the same result)

Answer 1

$a_0,a_1,a_2,....,a_n \\\text{ are terms of the convergent series};$

$S_n,S_{n+1}\text{ are partial sums}$

$S_n=a_0+a_1+...+a_{n-1}+a_{n}$

$S_{n+1}=a_0+a_1+...+a_{n}+a_{n+1}$

$a_n =S_{n+1}-S_n$

$\text{Since }a_0,a_1,a_2,....,a_n \\ \text{ are terms of the convergent series there is a limit of partial sums}$

$\lim\limits_{n\rightarrow\infin}S_n=S$

$\lim\limits_{n\rightarrow\infin}S_{n+1}=S$

$\lim\limits_{n\rightarrow\infin}a_n=\lim\limits_{n\rightarrow\infin}(S_{n+1}-S_n)=\lim\limits_{n\rightarrow\infin}S_{n+1}-\lim\limits_{n\rightarrow\infin}S_n=0$