I need some urgent help in determining whether the following infinite series actually converges or not. Upon graphing, it does seem like so but I can't be sure. I'm not concerned with evaluating this sum at the moment. I tried some techniques on http://www.wikihow.com/Determine-Convergence-of-Infinite-Series but can't figure out for some and was indeterminable for others.
Essentially if the series converges, that means there's an asymptote and that asymptote is the sum.
Hint. One may use standard Taylor series expansions as $i \to \infty$, to obtain $$ \begin{align} &(2i+4)\tan\left(\frac{\pi }{2i+4}\right)+(2i+3)\tan\left(\frac{\pi }{2i+3}\right)=2 \pi +\frac{\pi ^3}{6 i^2}+O\left(\frac1{i^3}\right) \\\\ &\frac{\pi }{1-\frac{(i+2)\left(1-\sin\left(\frac{\pi (i+1)}{2i+4}\right)\right)^2}{2}}=\pi +O\left(\frac1{i^3}\right) \\\\ &\frac{\pi }{1-(2i+3)\left(\frac{1-\sin\left(\frac{\pi (2i+1)}{2(2i+3)}\right)}{1+\sin\left(\frac{\pi (2i+1)}{2(2i+3)}\right)}\right)^2}=\pi +O\left(\frac1{i^3}\right) \end{align} $$ giving, as $i \to \infty$, a general term such that $$ u_i=\frac{\pi ^3}{6 i^2}+O\left(\frac1{i^3}\right) $$ and the given series is convergent by the comparison test.