Question

Using the integral test, how do you show whether `sum 1/n^3` diverges or converges from n=1 to infinity?

Answer 1

Check that `1/x^3` is (eventually) decreasing, then see whether `int_1^oo 1/(x^3) dx` diverges or converges.

Explanation:

If your grader or teacher won't let you claim that it is obvious that `1/x^3` is a decreasing function on `[1, oo]`, then

point out that for `f(x) = 1/x^3`, we get `f'(x) = -3/x^4` which is always negative, so `f` is always decreasing.

`int_1^oo 1/(x^3) dx = lim_(brarroo) int_1^b 1/(x^3) dx`

`= lim_(brarroo) (-1/(2x^2)]_1^b)`

`= lim_(brarroo) (-1/(2b^2) + 1/2)`

`= 1/2`

The integral converges, therefore the series converges. (We have not found what the series converges to.)