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.

The generalized binomial theorem says that
$$
(1+x)^{1/2}=\sum_{k=0}^\infty\binom{1/2}{k}x^k
$$
where $\binom{1/2}{0}=1$ and for $k\ge1$,
$$
\begin{align}
\binom{1/2}{k}
&=\frac{\frac12(\frac12-1)(\frac12-2)\cdots(\frac12-k+1)}{k!}\\
&=\frac{(-1)^{k-1}}{2^kk!}1\cdot3\cdot5\cdots(2k-3)\\
&=\frac{(-1)^{k-1}}{2^kk!}\frac{(2k-2)!}{2^{k-1}(k-1)!}\\
&=\frac{(-1)^{k-1}}{k2^{2k-1}}\binom{2k-2}{k-1}
\end{align}
$$
Thus,
$$
\begin{align}
(1+x)^{1/2}
&=1-\sum_{k=1}^\infty\frac2k\binom{2k-2}{k-1}\left(-\frac x4\right)^k\\
&=1-\sum_{k=0}^\infty\frac2{k+1}\binom{2k}{k}\left(-\frac x4\right)^{k+1}
\end{align}
$$
