<s>[INST] Do I need to use partial fractions to find $\frac{2n+1}{n^2(n+1)^2} = \frac{1}{n^2}-\frac{1}{(n+1)^2}$? I need to simplify $\frac{2n+1}{n^2(n+1^2)}$ as part of an exam question. The solution states $$\frac{2n+1}{n^2(n+1)^2} = \frac{1}{n^2}-\frac{1}{(n+1)^2}$$ 
In the solution it does not state how this simplification was made, I figured this could be done in quite a long winded fashion, using partial fractions. But from how it's written in the solutions it seems like this should be an easy simplification.
Is there a simple trick to simplifying fractions like this?
 [/INST] Note that $$\frac{2n+1}{n^2(n+1)^2} = \frac{(n^2 + 2n + 1)-n^2}{n^2(n+1)^2}$$
$$=\frac{(n+1)^2-n^2}{n^2(n+1)^2}$$
$$=\frac{(n+1)^2}{n^2(n+1)^2}-\frac{n^2}{n^2(n+1)^2}$$
$$=\frac1{n^2}-\frac1{(n+1)^2}$$
as desired. This approach is motivated by noticing that the original numerator is "almost" a recognizable perfect square, so one is tempted to add and subtract the missing quantity that completes the square.
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