# Til the last breath

Algebra Level 5

$\large \displaystyle \sum_{n=1}^{\infty} \dfrac{n(n+2)}{1\cdot2\cdot3+2\cdot3\cdot4+\cdots +n(n+1)(n+2)}$

If the above expression is in the form $$\color{red}{ \dfrac{A}{B}}$$ where $$\color{red}{A}$$ and $$\color{red}{B}$$ are co-prime positive integers, then let $$\color{darkgreen}{\phi}=\color{red}{A}\times \color{red}{B}$$.

$\large \color{purple}{\varphi}=\sqrt{\color{darkgreen}{\phi}\sqrt{\color{darkgreen}{\phi}^2\sqrt{\color{darkgreen}{\phi}^3\sqrt{\color{darkgreen}{\phi}^4\cdots}}}}$

Find the value of $$\color{blue}{\left(\dfrac{\sqrt{\color{purple}{\varphi}}}{5}\right)!}$$.

Notation:

$$!$$ denotes the factorial notation. For example, $$10! = 1\times2\times3\times\cdots\times10$$.

###### Inspired by Rohit Udaiwal.

Try the Part-2 here

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