# Keep it up-2

Algebra Level 5

$\displaystyle \sum_{\psi =1}^n (x+\psi-1)(x+\psi)=\phi$

If the equation above has $$\alpha$$ and $$\alpha+1$$ as its roots, then we get the relation between $$n$$ and $$\phi$$ as below. $n^2=1+\dfrac{\mathfrak{\color{blue}{T}}\phi}{n}$

Then find the value of the expression below.

$\large \sum_{i=\mathfrak{\color{blue}{\frac{\mathfrak{T}}{6}}}}^{\infty}\dfrac{40i}{i^4+4}$

Details

$$\alpha\in \mathbb{R} \text{ and } n,\mathfrak{T}\in\mathbb{Z}, n>1$$

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