The series in question:
$$\frac{5}{7^2+11^2} + \frac{9}{11^2+15^2} + \frac{13}{15^2+19^2} + \dots$$
Or in a concise form:
$$\sum_{i=1}^\infty {\frac{4i+1}{(4i+3)^2+(4i+7)^2}}$$
I tried to solve, and find a closed form of the above summation but got no luck. The denominator could not be factorised and decomposed and I couldn't transform the series into a telescopic one to solve it either.
I asked this to my maths professor and he looked at the series in question for a while, and declared it as a diverging one, so it can't be solved. He looked unsure.
Was he right? Is it a divergent series? If not, how can I solve it, if I can? Thanks!
Heuristically, the summand is the ratio of a linear polynomial to a quadratic polynomial, and so it grows similarly to the series $\sum_{i=1}^\infty \frac 1 i$, which diverges. This tells us that the original sum diverges as well. To show this, note that for sufficiently large $i$ ($i>80$, to be exact) we have $$\frac{4i+1}{(4i+3)^2+(4i+7)^2} = \frac{4i+1}{32i^2+80i+58} \geq \frac{4i}{33i^2} = \frac{4}{33}\frac 1 i$$ Now you can use the comparison test.