Taylor series of $\sqrt{1+x}$ using sigma notation I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation
I got till 
$1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on.
But I don't know what will come in sigma notation.

allow me to take one more tiny little step forward based on your excellent calculations.
$$
\sqrt{1+x}=\sum _{n=0}^{\infty } \frac{ 1}{(1-2 n)}
\frac{(2 n)!}{n! n!}\left(-\frac{x}{4}\right)^n
$$
or
$$
\sqrt{1+x}=\sum _{n=0}^{\infty } \frac{ C_{2n}^n}{(1-2 n)}
\left(-\frac{x}{4}\right)^n
$$
