identify the function represented by the following power series. \n∑(k = 0 to ∞) (-1)^k x^k / 12^k \nclick…

identify the function represented by the following power series. \n∑(k = 0 to ∞) (-1)^k x^k / 12^k \nclick the icon to view a table of taylor series for common functions. \nf(x) =

identify the function represented by the following power series. \n∑(k = 0 to ∞) (-1)^k x^k / 12^k \nclick the icon to view a table of taylor series for common functions. \nf(x) =

Answer

Explanation:

Step1: Recall geometric - series formula

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

Step2: Rewrite the given series

The given series $\sum_{k = 0}^{\infty}(- 1)^{k}\frac{x^{k}}{12^{k}}=\sum_{k = 0}^{\infty}\left(-\frac{x}{12}\right)^{k}$.

Step3: Identify the function

Comparing $\sum_{k = 0}^{\infty}\left(-\frac{x}{12}\right)^{k}$ with $\sum_{k = 0}^{\infty}r^{k}$, we have $r =-\frac{x}{12}$. So the function is $f(x)=\frac{1}{1+\frac{x}{12}}=\frac{12}{12 + x}$, for $\left|-\frac{x}{12}\right|\lt1$ (i.e., $|x|\lt12$).

Answer:

$\frac{12}{12 + x}$