Question

Using the integral test, how do you show whether `sum (n + 2) / (n + 1)` diverges or converges from n=1 to infinity?

Answer 1

The series diverges.

Explanation:

Without the integral test:

The terms do not go to `0` as `nrarroo` (they go to `1`), so the series diverges.

If we must use the integral test let's do so as quickly as possible.

`f(x) = (x+2)/(x+1) = (x+1+1)/(x+1) = 1+1/(x+1)`

`f'(x) = -1/(x+1)^2` which is negative, so `f` is a decreasing function.

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

`= lim_(brarroo) (x+ln abs(x+1))]_1^b`

`= lim_(brarroo)[ (b+ln(b+1)) - (1+ln2)]`

`= oo + oo -1-ln2 = oo`

The series diverges.