Question

How do you show that the series `1/2+2/3+3/4+...+n/(n+1)+...` diverges?

Answer 1

Is divergent.

Explanation:

`1/n < n/(n+1)` and

`sum_(k=1)^oo 1/k < sum_(k=1)^oo k/(k+1)`

but `sum_(k=1)^oo 1/k` is the so called harmonic series which is divergent. So `sum_(k=1)^oo k/(k+1)` is also divergent.