Gamma Under Sum

Calculus Level 5

\[S_x = \large \sum_{k = 0}^{\infty} \left [ {(-1)^k \times k \times 7^k \times \dfrac{x^{k+1}}{(e^x -1) \times \Gamma(k+2)}} \right ] \]

Given an equation above, let \[I = \int_{0}^{\infty} {S_x \, dx}\]

If \(I\) can also be expressed as \[I = \int_{0}^{1}{\dfrac{t^A}{t-1} (\ln(t))^B \, dt}, \]

find \(A+B\).

This is an Original Problem.
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