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

May 27, 2015

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

$f \left(x\right) = \frac{1}{{x}^{2} + 1}$

and $f ' \left(x\right) = - \frac{2 x}{{x}^{2} + 1} ^ 2$

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 ${\int}_{1}^{\infty} \frac{1}{{x}^{2} + 1}$

of which we can see that it is $= {\lim}_{t = \infty} {\left[{\tan}^{-} 1 \left(x\right)\right]}_{1}^{t}$

$= {\lim}_{t = \infty} {\tan}^{-} 1 \left(t\right) - {\tan}^{-} 1 \left(1\right)$

$= \frac{\pi}{2} - \frac{\pi}{4}$

$= \frac{\pi}{4}$

Thus by the integral test, our series is convergent