nth term test for the series $\sum_{n=1}^{\infty }{1/n} $

Question

the n-th term test for the series $\sum_{n=1}^{\infty }{1/n}$ implies that the series converges since $\lim_{n\to\infty} 1/n =0$
but the series actually diverges. what is the error in the procedure followed?

Answer 1

Be careful: the theorem states

If $\sum a_n$ converges, then $\lim_{n\to \infty}{a_n}=0$

which conversely means

If $\lim_{n\to \infty}{a_n}\neq 0$ then $\sum a_n$ does not converge.

This means that $\lim_{n\to \infty}{a_n}=0$ is a necessary but not sufficient condition for a series to converge.

Answer 2

If the series $\sum_{n=1}^{\infty} a_n$ converges, than limit of $n$-th term $a_n$ is zero as $n\to\infty$. It's because convergence means existence of $\lim_{n\to\infty} S_n$, where $S_n = \sum_{k=1}^n a_k$, and $a_n = S_{n} - S_{n-1}$, so $\lim_{n\to\infty} a_n = \lim_{n\to\infty} S_n - \lim_{n\to\infty} S_{n-1} = 0$ as latter two limits are equal. However, converse isn’t true as your example shows.