<s>[INST] Induction proof of a series Suppose we have the series 
$$f(n) = \sum_{i=1}^n \frac{2i-1}{i^4 - 2 i^3 + 3 i^2 - 2 i + 2}.$$
As a hint it was said that this series "telescopes". I observed the pattern to be $f(n)=\frac{n^2}{n^2 +1}$. I wish to prove this via induction. The base case holds, since $f(1)=\frac{1}{2}$ which comes out of both the definition of $f(n)$ and my proposed formula. Now I wish to prove that for some arbitrary $k$ we assume $f(k)=\frac{k^2}{k^2+1}$, then I wish to prove that $f(k+1)=\frac{(k+1)^2}{(k+1)^2+1} (=\frac{k^2 +2k +1}{k^2 +2k +2})$.
I wanted to start out by noticing: 
$$f(k+1) = \sum_{i=1}^{k+1} \frac{2i-1}{i^4 - 2 i^3 + 3 i^2 - 2 i + 2}= \\f (k)+ \frac{2(k+1)-1}{(k+1)^4 - 2 (k+1)^3 + 3 (k+1)^2 - 2 (k+1) + 2}$$
This eventually simplifies to
$$ f(n+1)= \frac{k^2}{k^2 +1} + \frac{2k+1}{k^4 +2k^3+3k^2+2k+2}$$
 [/INST] Note that$$\frac{i^2}{i^2+1}-\frac{(i-1)^2}{(i-1)^2+1}=\frac{2i-1}{i_4-2i^3+3i^2-2i+2}$$and that therefore\begin{align}\sum_{i=1}^n\frac{2i-1}{i^4-2i^3+3i^2-2i+2}&=\sum_{i=1}^n\frac{i^2}{i^2+1}-\frac{(i-1)^2}{(i-1)^2+1}\\&=\frac{n^2}{n^2+1}-\frac{0^2}{0^2+1}\\&=\frac{n^2}{n^2+1}.\end{align}
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