Determining whether a series is convergent

Question

If $a_n := \frac{n+1}{n}$, how can I determine whether $\sum_{n=1}^{\infty} a_n$ converges? I know that

$$\lim_{n\to \infty} \frac{n+1}{n} = 1$$

But the expression for the partial sum of $n$ terms is

$$\sum_{i=1}^{n} \frac{i+1}{i} = n + \sum_{i=1}^{n} \frac{1}{i}$$

and I don't know how to evaluate that last summand.

Answer 1

All you need is that the last summand is positive, so the sum exceeds $n$ and so diverges.