Question

Determine whether the series$\sum _{k=0}^{\infty }\frac{5^{k+1}}{{\left(-6\right)}^k}$ converges or diverges. Give the sum of the convergent series.

Short answer

The series $\sum _{k=0}^{\infty }\frac{5^{k+1}}{{\left(-6\right)}^k}$converges to $\frac{30}{11}$.

Step-by-step solution

Step 1. Given information.

Given a series $\sum _{k=0}^{\infty }\frac{5^{k+1}}{{\left(-6\right)}^k}$.

Step 2. Find if the series converges or not.

The series$\sum _{k=0}^{\infty }\frac{5^{k+1}}{{\left(-6\right)}^k}$can be expressed as$\underset{k=0}{\overset{\infty }{\sum 5}}{\left(\frac{-5}{6}\right)}^k$.

The series $\sum _{k=0}^{\infty }\frac{5^{k+1}}{{\left(-6\right)}^k}$is in the standard form $\sum _{k=0}^{\infty }c{r}^k$for a geometric series with $c=5$and $r=-\frac{5}{6}$.

The geometric series converges if and only if $\left|r\right|<1$.

Since localid="1648983652785" $r=-\frac{5}{6}$, it follows that the series converges.

Step 3. Find the value to which the series converges.

If the geometric series $\sum _{k=0}^{\infty }c{r}^k$converges, it converges to $\frac{c}{1-r}$.

So, the series localid="1648983661968" $\sum _{k=0}^{\infty }\frac{5^{k+1}}{{\left(-6\right)}^k}$converges to localid="1648983659158" $\frac{5}{1-\left(-{\displaystyle \frac{5}{6}}\right)}$, that is localid="1648983666395" $\frac{30}{11}$.