A Weird Infinity Fractional Sequence ~ Inspired by Sourasish K

1 2 + 3 2 4 + 3 5 2 4 8 + 3 5 7 2 4 8 12 + = ? \begin{aligned} \dfrac{1}{2} + \dfrac{3}{2 \cdot 4} + \dfrac{3 \cdot 5 }{2 \cdot 4 \cdot 8 } + \dfrac{3 \cdot 5 \cdot 7 }{ 2 \cdot 4 \cdot 8 \cdot 12 } + \cdots = \text{?} \end{aligned}

Clarification: The n th n^\text{th} term of this series can be expressed as n 2 3 n 1 ( 2 n n ) \dfrac{n}{2^{3n - 1}} \dbinom{2n}{n} .


Inspiration .

Try another problem on my set Let's Practice !
2 2 2 \sqrt{2} 2 e 2^{e} e 2 e^{2}

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1 solution

Ankit Kumar Jain
May 22, 2017

( 1 x ) n = 1 + n x + n ( n + 1 ) 2 ! x 2 + + n ( n + 1 ) ( n + 2 ) ( n + r 1 ) r ! x r + terms (1-x)^{-n}=1+nx+\dfrac{n(n+1)}{2!}x^2+\cdots+\dfrac{n(n+1)(n+2)\cdots (n+r-1)}{r!}x^{r}+\cdots\infty\quad\text{terms}


Substitute x = 1 2 , n = 3 2 \boxed{x=\dfrac12 , n = \dfrac32}

1 + 3 4 + 3 5 4 8 + 3 5 7 4 8 12 + = 2 2 \Rightarrow 1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots=2\sqrt{2}

Multiplying throughout by 1 2 \dfrac12 ,

1 2 + 3 2 4 + 3 5 2 4 8 + 3 5 7 2 4 8 12 + = 2 \Rightarrow \dfrac12+\dfrac3{2\cdot4}+\dfrac{3\cdot5}{2\cdot4\cdot8}+\dfrac{3\cdot5\cdot7}{2\cdot4\cdot8\cdot12}+\cdots = \boxed{\sqrt{2}}

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