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

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Answer 1 · 2015-05-27T17:17:40

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

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