Question

In Problems 61–70, express each sum using summation notation.

$\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-........+{\left(-1\right)}^{12}{\left(\frac{2}{3}\right)}^{11}$

Short answer

The required summation notation is $\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-........+{\left(-1\right)}^{12}{\left(\frac{2}{3}\right)}^{11}=\sum _{k=1}^{11}{\left(-1\right)}^{n+1}{\left(\frac{2}{3}\right)}^k$.

Step-by-step solution

Step 1. Write the given information.

The given sum is:

$\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-........+{\left(-1\right)}^{12}{\left(\frac{2}{3}\right)}^{11}$

Step 2. Write the sum using summation notation.

Write each term in the same format:

$$\begin{gathered} \frac{2}{3}-\frac{4}{9}+\frac{8}{27}-........+{\left(-1\right)}^{12}{\left(\frac{2}{3}\right)}^{11} \\ ={\left(-1\right)}^2{\left(\frac{2}{3}\right)}^1+{\left(-1\right)}^3{\left(\frac{2}{3}\right)}^2+{\left(-1\right)}^4{\left(\frac{2}{3}\right)}^3+.......+{\left(-1\right)}^{12}{\left(\frac{2}{3}\right)}^{11} \\ ={\left(-1\right)}^{1+1}{\left(\frac{2}{3}\right)}^1+{\left(-1\right)}^{2+1}{\left(\frac{2}{3}\right)}^2+{\left(-1\right)}^{3+1}{\left(\frac{2}{3}\right)}^3+.......+{\left(-1\right)}^{11+1}{\left(\frac{2}{3}\right)}^{11} \\ =\sum _{k=1}^{11}{\left(-1\right)}^{n+1}{\left(\frac{2}{3}\right)}^k \end{gathered}$$

Therefore,role="math" localid="1646798011820" $\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-........+{\left(-1\right)}^{12}{\left(\frac{2}{3}\right)}^{11}=\sum _{k=1}^{11}{\left(-1\right)}^{n+1}{\left(\frac{2}{3}\right)}^k$.