Ratio of Diverging Series'!

Calculus Level 5

\[\mathcal P=\sum_{n=1}^{\infty}\dfrac{3+4+5+\cdots+(\text{n terms})}{1\cdot 3\cdot 5+2\cdot 4\cdot 6+3\cdot 5\cdot 7+\cdots +(\text{n terms})}\]

If the value of \(\mathcal P\) can be written in the form \(\color{red}{\dfrac AB}\) , where \(A\) and \(B\) are coprime positive integers, find \(\sqrt{\color{red}{A}+\color{red}{B}+5}\).

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