Pattern for Induction?

Question

Find and prove a formula for the sum $$\frac{1^3}{1^4+4} - \frac{3^3}{3^4+4}+...+\frac{(-1)^n(2n+1)^3}{(2n+1)^4 + 4}$$ where $n$ is an integer.

I tried listing out the partial sums of the sequence to see if there was a pattern, however the only thing I can make out is that the denominators of the sum always either end in 5 or 9.

I would just like a hint as to how to solve this. Thanks.

Answer 1

Extended hint:

Writing down 4-5 first partial sums reveals after reducing the fractions the general pattern: $$ S_n=(-1)^n\frac{n+1}{4(n+1)^2+1}. $$ It remains to apply induction or telescoping. The following facts will be useful: $$\begin{array}{ll} (2n+1)^4+4&=(4n^2+1)(4n^2+8n+5),\\ 4n^2+8n+5&=4(n+1)^2+1,\\ (2n+1)^3-n(4n^2+8n+1)&=(n+1)(4n^2+1). \end{array} $$

$\frac{(2n+1)^3}{(2n+1)^4+4}=\frac{n}{4n^2+1\vphantom{)^2}}+\frac{n+1}{4(n+1)^2+1}.$