$\lim_{n\rightarrow\infty}\left(\dfrac{1^{2016}+2^{2016}+\cdots+n^{2016}}{n^{2017}}\right)=\color{orange}{\psi}\lim_{n\rightarrow\infty}\left(\dfrac{1^{2017}+2^{2017}+\cdots+n^{2017}}{n^{2018}}\right)$ If above equation holds true we get $\color{orange}{\psi}=1+\dfrac{1}{\color{indigo}{\beta}}$ such that $2(\color{indigo}{\beta}-1)=\color{forestgreen}{\phi}(\color{forestgreen}{\phi}+1)$.

$\Large\mathfrak{P}=\sum_{n=\sqrt[3]{\color{forestgreen}{\phi+1}}}^{\color{forestgreen}{\phi}}\left(\dfrac{1}{n^2+3n+2}\right)$ If $\mathfrak{P}$ can be represented in the form $\color{red}{\dfrac{\delta}{\alpha}}$, then find $\color{red}{\dfrac{\alpha}{\delta+1}}$.

$\textbf{Details:}$

$\psi\in\mathbb R ~~;\beta,\phi,\alpha,\delta \in\mathbb{Z}$.

$\delta$ and $\alpha$ are coprime.

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