Til the last breath

Algebra Level 5

n=1n(n+2)123+234++n(n+1)(n+2) \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 AB\color{#D61F06}{ \dfrac{A}{B}} where A\color{#D61F06}{A} and B\color{#D61F06}{B} are co-prime positive integers, then let ϕ=A×B\color{#456461}{\phi}=\color{#D61F06}{A}\times \color{#D61F06}{B}.

φ=ϕϕ2ϕ3ϕ4\large \color{#69047E}{\varphi}=\sqrt{\color{#456461}{\phi}\sqrt{\color{#456461}{\phi}^2\sqrt{\color{#456461}{\phi}^3\sqrt{\color{#456461}{\phi}^4\cdots}}}}

Find the value of (φ5)!\color{#3D99F6}{\left(\dfrac{\sqrt{\color{#69047E}{\varphi}}}{5}\right)!}.

Notation:

!! denotes the factorial notation. For example, 10!=1×2×3××1010! = 1\times2\times3\times\cdots\times10 .


Inspired by Rohit Udaiwal.

Try the Part-2 here

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