Limit of infinite sequence from partial sum

Question

I think there was a rule in Calculus that mentions this, but I am not sure.

If I need to find $\lim_{n \to \infty} a_n$ and I am only given the nth partial sum: $S_n =\sum_{k=1}^{n} a_k = f(n)$

To find $\lim_{n \to \infty} a_n$ I just have to find $\lim_{n \to \infty} f(n)$ correct?

Answer 1

As noticed by lulu in the comment note that

$$S_n-S_{n-1} =\sum_{k=1}^{n} a_k-\sum_{k=1}^{n-1} a_k = a_n\color{red}{+\sum_{k=1}^{n-1} a_k-\sum_{k=1}^{n-1} a_k}=a_n=f(n)-f(n-1)$$

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Remark:

• that is precisely the reason for which $a_n\to 0$ is a necessary condition for the convergence of any series $\sum_{k=1}^{\infty} a_k$, indeed

$$\lim_{n\to \infty}S_n=\sum_{k=1}^{\infty} a_k=L \implies S_n-S_{n-1} =a_n \to 0$$