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

1 Answer
May 27, 2015

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