Integral Test

I have to use the integral test to figure out if the summation of 1/(n²+4) converges or diverges.

the first thing I did was set it to a function (1/(x²+4)) then I set up an integral from 1 to infinity for that function. Then took the limit as b>infinity from 1 to b for f(x).

I'm stuck here. The solution manual says that I should get the following:

lim as b>inf [(1/2)Tan^-1(x/2)] from b to 1.

Can someone explain this to me please? I can figure out the steps after but I don't understand this step.

Hey bakinbacon.

You need to consider the integral which should be [(1/2)arctan(x/2)] {x=1,x=infinity} and show that the answer is finite and if so then the series converges by the integral test.

If it doesn't converge then the series will diverge.

Which "step" are you talking about?
Are you okay with [itex]\inti\frac{dx}{a^2+ x^2}= arctan(ax)+ C[/tex]?

Do you know that tan(x) goes to infinity as x goes to ? From that, arctan(x) goes to as x goes to infinity.