Question

Using the integral test, how do you show whether `sum 1 / [sqrt(n) * (sqrt(n) + 1)]` diverges or converges from n=1 to infinity?

Answer 1

The sum is unbounded

Explanation:

We know

`sum_{i=1}^{n} 1 / [sqrt(i) * (sqrt(i) + 1)]geint_{1}^n(dx)/(sqrt(x)(sqrt(x)+1)} = 2log_e(1+sqrt(n))`

because `1 / [sqrt(i) * (sqrt(i) + 1)]` is monotonic decreasing.

but `lim_{n->oo}2log_e(1+sqrt(n)) = oo`

so the sum is unbounded

Attached a figure with the comparisson between series and integral.