Question

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\left(\begin{array}{c}-\frac{1}{2}\\ n\end{array}\right){\left(-\frac{1}{2}\right)}^{\mathbf{n}}$.

Short answer

The sum of the series, i.e. $\sum _{n=0}^{\infty }\left(\begin{array}{c}-\frac{1}{2}\\ n\end{array}\right){\left(-\frac{1}{2}\right)}^n=\sqrt{2}$,

Step-by-step solution

Given Information

The given series, i.e. $\sum _{n=0}^{\infty }\left(\begin{array}{c}-\frac{1}{2}\\ n\end{array}\right){\left(-\frac{1}{2}\right)}^n$,

Definition of Sum of a Series

The total of all the terms in a sequence can be combined and written as the sum of a series.

Find the sum of the series

Use the Maclaurin series of${\left(1+x\right)}^n$,

${(1+x)}^p=\sum _{n=0}^{\infty }\left(\begin{array}{c}p\\ n\end{array}\right){\left(x\right)}^n$

Further, put $x=\frac{1}{2}$and simplify as:

$$\begin{gathered} {\left(1-\frac{1}{2}\right)}^{-1/2}=\sum _{n=0}^{\infty }\left(\begin{array}{c}-\frac{1}{2}\\ n\end{array}\right){\left(-\frac{1}{2}\right)}^n \\ \sum _{n=0}^{\infty }\left(\begin{array}{c}-\frac{1}{2}\\ n\end{array}\right){\left(-\frac{1}{2}\right)}^n=\sqrt{2} \end{gathered}$$

Hence, the sum of the series, i.e., $\sum _{n=0}^{\infty }\left(\begin{array}{c}-\frac{1}{2}\\ n\end{array}\right){\left(-\frac{1}{2}\right)}^n=\sqrt{2}$.