Establish the convergence or divergence of a sequence Establish the convergence or divergence of the sequence (y_n), where:
y_n := 1/(n+1) + 1/(n+2) + ... = 1/(2n) for n /in N.

You can use the monotone convergence theorem:
$$y_{n+1}-y_n=\frac1{2n+1}+\frac1{2n+2}-\frac1{n+1}=\frac1{(2n+1)(2n+2)}>0 $$
so the sequence $(y_n)$ is increasing.
It is bounded from above: since for all $k\ge 1$, $\dfrac1{n+k}\le \dfrac1{n+1}$, we have:
$$y_n=\sum_1^n\frac1{n+k}\le \frac n{n+k} <1.$$
