Q:

Negative convergent series intuition

I am interested on the behavior of the series $$\lim_{n \to \infty} \sum_{k=1}^n a_k < 0 $$ with $\lim_{n \to \infty} a_n = a^* $. Can we say that this series converges? Also, is it possible to define any condition on $a_k$ (e.g., decreasing sequence) in order to conclude that $a_k <0$? Sorry for my naive question I am not experienced with series. 

A:

When you write $\lim_{n \to \infty} \sum_{k=1}^n a_k < 0$ (or, as it is usually written, $\sum_{k=1}^\infty a_k < 0$, you're assuming that the limit exists. In other words, the sum indeed converges. One can show that if the sum converges, then $\lim_{n \to \infty} a_n = 0$ necessarily. 
However, the converse is not true, i.e. to $\lim_{n \to \infty} a_n = 0$, then $\sum_{k=1}^\infty a_k$ may or may not converge. The most classic and famous example is the harmonic series, where $a_n = \frac{1}{n}$. Clearly $a_n = \frac{1}{n} \to 0$, but $\sum_{k=1}^\infty \frac{1}{n}$ diverges to $+\infty$.

