identify the function represented by the following power series. \n sum_{k = 0}^{infty}(-1)^{k}\frac{x^{k +…

identify the function represented by the following power series. \n sum_{k = 0}^{infty}(-1)^{k}\frac{x^{k + 1}}{5^{k}} \nclick the icon to view a table of taylor series for common functions.\n\nf(x)=

identify the function represented by the following power series. \n sum_{k = 0}^{infty}(-1)^{k}\frac{x^{k + 1}}{5^{k}} \nclick the icon to view a table of taylor series for common functions.\n\nf(x)=

Answer

Explanation:

Step1: Rewrite the power - series

We have $\sum_{k = 0}^{\infty}(-1)^{k}\frac{x^{k + 1}}{5^{k}}=x\sum_{k = 0}^{\infty}(-1)^{k}(\frac{x}{5})^{k}$.

Step2: Recall the geometric series formula

The geometric series formula is $\sum_{k = 0}^{\infty}r^{k}=\frac{1}{1 - r}$, where $|r|\lt1$. Here $r=-\frac{x}{5}$.

Step3: Substitute $r$ into the formula

Since $x\sum_{k = 0}^{\infty}(-1)^{k}(\frac{x}{5})^{k}=x\sum_{k = 0}^{\infty}(-\frac{x}{5})^{k}$, and using the geometric - series formula, we get $x\cdot\frac{1}{1-(-\frac{x}{5})}$.

Step4: Simplify the expression

$x\cdot\frac{1}{1+\frac{x}{5}}=\frac{x}{\frac{5 + x}{5}}=\frac{5x}{5 + x}$, for $|-\frac{x}{5}|\lt1$ (i.e., $|x|\lt5$).

Answer:

$\frac{5x}{x + 5}$