limit of the sum $ \sum_{i=1}^N a_N(i) $

Question

I have that strange sum: $$ \sum_{i=1}^N a_N(i) $$ for which I want to compute the limit as $N\to \infty$, except that I have as an additional condition $\lim_{N\to \infty}a_N(i)=0$.

I have that bizzare intuition that this serie should converge to zero except that I can't figure how ? or maybe my intuition is wrong !! What do you think ?

Answer 1

The series need not converge to zero, if all you know is that $\lim_{N\to\infty}a_N(i)=0$. Take, for instance, the degenerate case $$ a_N(i) = \frac1N,\qquad\text{for all $i=1,\ldots N$}. $$ Then $\sum_{i=1}^N a_N(i)$ is equal to $1$ for every $N$.

Answer 2

In order to make this conclusion you need additional conditions. A popular one is that there is some sequence $b_i \ge 0$ such that $\sum_{i=1}^\infty b_i < \infty$ and $|a_N(i)| \le b_i$ for all $i$ and $N$.