Show that $\sum\limits_{n=0}^{\infty }{2n\choose n}\frac{1}{8^n}=\sqrt {2}$

Question

$$\sum _{n=0}^{\infty }\:\begin{pmatrix}2n\\ n\end{pmatrix}\:\frac{1}{8^n}=\sqrt{2}$$ I know that I should use some Taylor series, but which one I'm not sure.

You can get genuine binomial coefficients using either {n \choose k} or \binom{n}{k}. — Brian M. Scott, Nov 18 '16 at 19:55

score 0 · Answer 1 · answered Nov 18 '16 at 19:53

0

Well, it is known that

$$\left ( 1-4 x^2 \right )^{-1/2} = \sum_{n=0}^{\infty} \binom{2 n}{n} x^{2 n} $$

Plug in $x^2=\frac18$.

answered Nov 18 '16 at 19:53

Ron Gordon

138,521

Show that $\sum\limits_{n=0}^{\infty }{2n\choose n}\frac{1}{8^n}=\sqrt {2}$

1 Answers1