identify the function represented by the following power series. \n∑(k = 0 to ∞) (-1)^k * (x^(3k))/(6^k)…

Explanation:

Step1: Rewrite the power - series

We can rewrite the given power series $\sum_{k = 0}^{\infty}(-1)^{k}\frac{x^{3k}}{6^{k}}$ as $\sum_{k = 0}^{\infty}(-1)^{k}(\frac{x^{3}}{6})^{k}$.

Step2: Recall the geometric - series formula

The geometric series formula is $\sum_{k = 0}^{\infty}r^{k}=\frac{1}{1 - r}$, where $|r|\lt1$. In our case, $r=-\frac{x^{3}}{6}$.

Step3: Substitute $r$ into the formula

Substituting $r =-\frac{x^{3}}{6}$ into the geometric - series formula, we get $\frac{1}{1-(-\frac{x^{3}}{6})}=\frac{1}{1+\frac{x^{3}}{6}}=\frac{6}{6 + x^{3}}$.

Answer:

$\frac{6}{6 + x^{3}}$