Sum of ${1 - {(1/8)} + {(1×3)}/{(8×16)} - {(1×3×5)}/{(8×16×24)} + .... }$

Question

Sum of the following series (till infinite terms):

${1 - {(1/8)} + {(1×3)}/{(8×16)} - {(1×3×5)}/{(8×16×24)} + .... }$

I tried writing the general term and then proceeding... but its not helping. Can anyone help me out? Thanks in advance!!

Answer 1

$$1\times2\times3\times\dots\times n=n!$$

$$2\times4\times6\times\dots\times2n=2^nn!$$

$$1\times3\times5\times\dots\times(2n-1)=\frac{(2n)!}{2^nn!}$$

So, your sum can be written as

$$S=\sum_{n=0}^\infty\frac{(2n)!}{(-16)^n(n!)^2}=\sum_{n=0}^\infty\frac1{(-16)^n}\binom{2n}n=\frac1{\sqrt{1+\frac14}}=\frac2{\sqrt5}$$

which comes about due to the binomial expansion of $(1+4x)^{-1/2}$ and having $x=\frac1{16}$.

---

Update:

There just so happens to be a full Wikipedia on this:

https://en.wikipedia.org/wiki/Central_binomial_coefficient

Answer 2

This is equivalent to

$$S = 1 + \frac{1}{2}\sum_{n=1}^{n=\infty} (-1)^{n+1}\left[{{2n-1}\choose n}. (\frac{1}{4})^n . (\frac{1}{4})^{n-1}\right]$$

The one that you see in the square brackets looks like binomial

Could you take it from here. It is too late here in my place to manipulate this. Probably in the morning.

Goodluck