Question related to the sigma during expectation in probability

Question

How is this below possible:

$ \sum_{i=1}^\infty 1/i = \infty $

Answer 1

It stands:

$$\sum_{i=1}^{\infty} \frac{1}{i}=\infty \\ \\ \text{ AND } \\ \\ \lim_{n \to +\infty} \sum_{i=1}^{n} \frac{1}{i}=\infty$$

BUT NOT:

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

EDIT:

If the series $\sum_{n=1}^{\infty} \frac{1}{n}$ would converge,from the Cauchy criterion,there would be a $n_0 \in \mathbb{N}$ such that $\forall m>n \geq n_0:$ $$\frac{1}{n+1}+\frac{1}{n+2}+ \dots +\frac{1}{m}<\frac{1}{2}$$

Specifically:

$$\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{n+n}<\frac{1}{2}$$

But:

$$\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{n+n} \geq \frac{1}{n+n} +\frac{1}{n+n} \dots+\frac{1}{n+n} =\frac{n}{2n}=\frac{1}{2}$$

Therefore,the series diverges.