I am having trouble evaluating the limit of an infinite sum. The expression is:
$$\lim_{n\to\infty} \sum_{i=1}^{\infty} \exp\left(-\frac{n}{i}\right)$$
Now, for any fixed $n$, the terms in the tail of the sum goes to $\exp(0)=1$, so it clearly diverges. However, at the same time each term is going to $0$; If we actually "set" $n=\infty$, the sum is $0$. So, which one is the right approach? Is this limit actually defined?
No, the limit is not defined. For each $n\in\mathbb N$, the series $\sum_{i=1}^\infty\exp\left(-\frac ni\right)$ diverges, since we don't have $\lim_{i\to\infty}\exp\left(-\frac ni\right)=0$. Therefore, it doesn't even make sense to study that limit.