Question

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

Answer 1

Before using the integral test, you need to make sure that your function is decreasing, so we get:

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

and `f'(x) = -(2x)/(x^2 + 1)^2`

Which is negative for all `x > 0`

Thus our series is decreasing.

we also need to know that the function is always positive, which we can see that it is.

Then we can solve for `int_1^oo 1/(x^2 + 1)`

of which we can see that it is `=lim_(t=oo) [tan^-1(x)]_1^t`

`= lim_(t=oo) tan^-1(t) - tan^-1(1)`

`= pi/2 - pi/4`

`=pi/4`

Thus by the integral test, our series is convergent