The notation $\sum_{n=0}^{+\infty} a_n$ just means the limit $\lim_{k \to \infty} \sum_{n=0}^k a_n$ and it's a way to make sense of expressions like $a_0 + a_1 + a_2 + \dots$, which are formally meaningless.
This is because a sum, and other operation as well, are only defined for a finite number of operand:
$$a_0 + a_1 + a_2 := (a_0 + a_1) + a_2 = a_0 + (a_1 + a_2)$$
If you manipulate directly a summation with infinitely many terms you can end up with the most surprising and strange results. See this for example.
In general you can't easily calculate this limit but for a telescoping sum, as you know, the terms cancel out leaving a handy expression:
$$
\sum_{n=1}^{k} \log \frac{n+1}{n} = \sum_{n=1}^{k} [\log (n+1) - \log n ] = \log 2 - \log 1 + \log 3 - \log 2 + \\
+\dots + \log(k-1+1)-\log(k-1) + \log(k+1)-\log k = \log (k+1)
$$
So the result is
$$
\sum_{n=1}^{+\infty} \log \frac{n+1}{n} = \lim_{k \to +\infty} \sum_{n=1}^{k} \log \frac{n+1}{n} = \lim_{k \to \infty} \log (k+1) = +\infty
$$