# Serious Series

**Calculus**Level 5

\[\large\mathfrak{T}(n)=\dfrac{0.75n^2}{1-2^2+3^3+4-5^2+6^3+\cdots \text{(3n terms)}}\]

\[\large\mathfrak{S}(n)=\dfrac{\frac{1}{\mathfrak{T}(n)}+13n+9}{9}\]

\[\large\mathfrak{I}=\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{1}{\mathfrak{S}(n)}\right)=\dfrac{\pi^{\alpha}}{\beta}\]

\[\large\mathfrak{S}(\alpha!+\beta)=\, ?\]

**Details and Assumptions**

- \(n,\alpha,\beta\in\mathbb Z\) and \(n\geq 1\)