
Integral Test
I have to use the integral test to figure out if the summation of 1/(n^{2}+4) converges or diverges.
the first thing I did was set it to a function (1/(x^{2}+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.

Re: Integral Test
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.

Re: Integral Test
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 $\displaystyle \pi/2$? From that, arctan(x) goes to $\displaystyle \pi/2$ as x goes to infinity.