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Calculus Level 4

\[\lim_{n\rightarrow\infty}\left(\dfrac{1^{2016}+2^{2016}+\cdots+n^{2016}}{n^{2017}}\right)=\color{orange}{\psi}\lim_{n\rightarrow\infty}\left(\dfrac{1^{2017}+2^{2017}+\cdots+n^{2017}}{n^{2018}}\right)\] 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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