Question 2.40: Prove that if a series u1 + u2 + u3 +... is to converge, we ...

If S_n is the sum of the first n terms of the series, then S_{n+1}=S_n+u_n. Hence, if \lim _{n \rightarrow \infty} S_n exists and equals S, we have \lim _{n \rightarrow \infty} S_{n+1}=\lim _{n \rightarrow \infty} S_n+\lim _{n \rightarrow \infty} u_n or S=S+\lim _{n \rightarrow \infty} u_n,  i.e.,  \lim _{n \rightarrow \infty} u_n=0.
Conversely, however, if \lim _{n \rightarrow \infty} u_n=0, the series may or may not converge. See Problem 2.150.