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Hint: this is a telescoping series sum (I have no prior knowledge of partial fraction decomposition)

Attempt: I tried to complete the square but the numerator had an unsimplifiable term. So I couldn't find a pattern. I just need a hint on how to convert this into a telescoping series.

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    $\begingroup$ Recall Sophie Germain's identity $$x^4+4 = (x^4+4x^2+4)-(4x^2)=(x^2+2)^2-(2x)^2= (x^2+2x+2)(x^2-2x+2)$$ $\endgroup$ Commented Jul 13, 2020 at 16:33

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You say that you have not been able to see a patterm $$S_p=\sum_{n=1}^{p}\frac{(n^2-\frac 12)}{(n^4+\frac 14)}$$ generates the sequence $$\left\{\frac{2}{5},\frac{8}{13},\frac{18}{25},\frac{32}{41},\frac{50}{61},\cdots\right\}$$ The numerators seem to be $2p^2$.

Now, subtract $1$ from each denominator to have $$\left\{4,12,24,40,60,\cdots\right\}$$ which seem to be $2p(p+1)$.

So, if I am not wrong $$S_p=\frac{2p^2}{2p(p+1)+1}$$

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  • $\begingroup$ Thanks for the help, idk why I was missing this...any other approach other than observation? $\endgroup$
    – sfsg
    Commented Jul 13, 2020 at 16:00
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    $\begingroup$ @user807908. Taking into you constraints, I do not see. Moreover, I reacted to the fact that you said that you did not find any pattern. $\endgroup$ Commented Jul 14, 2020 at 1:45

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