So this is the problem: http://postimg.org/image/5g815zgk5/
I am getting $\lim_{b\to\infty} 2\sec^{-1}(2b) - 2\sec^{-1}2$
Now what? What do I do with $\sec^{-1}(2b)$? What happens to a trig function when the $b$ goes to infinity? What does it become?
Thank you!
Note that as $x$ approaches $\frac{\pi}{2}$ from the left, $\cos x\to 0$, and therefore $\sec x\to\infty$.
It follows that $\sec^{-1}(2b)\to\frac{\pi}{2}$ as $b\to\infty$. Thus $$\int_1^\infty \frac{dt}{t\sqrt{t^2-\frac{1}{4}}}$$ converges, and therefore by the Integral Test our series converges.
Remark: If the question specified that one must use the Integral Test, the task has been carried out. However, it is I think more natural to use the Limit Comparison Test, comparing with $\sum_1^\infty \frac{1}{n^2}$. (And then the usual proof that this series converges uses the Integral Test. However, by now the convergence of this series may be a standard fact that one can quote without proof.)