# As you like it

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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