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Calculus Level 4

limn(12016+22016++n2016n2017)=ψlimn(12017+22017++n2017n2018)\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 ψ=1+1β\color{orange}{\psi}=1+\dfrac{1}{\color{indigo}{\beta}} such that 2(β1)=ϕ(ϕ+1)2(\color{indigo}{\beta}-1)=\color{forestgreen}{\phi}(\color{forestgreen}{\phi}+1).

P=n=ϕ+13ϕ(1n2+3n+2)\Large\mathfrak{P}=\sum_{n=\sqrt[3]{\color{forestgreen}{\phi+1}}}^{\color{forestgreen}{\phi}}\left(\dfrac{1}{n^2+3n+2}\right) If P\mathfrak{P} can be represented in the form δα\color{red}{\dfrac{\delta}{\alpha}}, then find αδ+1\color{red}{\dfrac{\alpha}{\delta+1}}.


Details:\textbf{Details:}

ψR  ;β,ϕ,α,δZ\psi\in\mathbb R ~~;\beta,\phi,\alpha,\delta \in\mathbb{Z}.

δ\delta and α\alpha are coprime.

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