Question

Determine whether the series $\sum _{k=0}^{\infty }\frac{3}{2^k}$converges or diverges. Give the sum of the convergent series.

Short answer

The series$\sum _{k=0}^{\infty }\frac{3}{2^k}$converges to 6.

Step-by-step solution

Step 1. Given information.

Given a series$\sum _{k=0}^{\infty }\frac{3}{2^k}$.

Step 2. Find if the series converges or not.

The series $\sum _{k=0}^{\infty }\frac{3}{2^k}$is in the standard form localid="1648980136761" role="math" $\sum _{k=0}^{\infty }c{r}^k$for a geometric series with localid="1648886392804" $c=3$and $r=\frac{1}{2}$.

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

Since localid="1648886396247" $r=\frac{1}{2}$, it follows that the series$\sum _{k=0}^{\infty }\frac{3}{2^k}$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$\sum _{k=0}^{\infty }\frac{3}{2^k}$converges to$\frac{3}{1-{\displaystyle \frac{1}{2}}}$, that is 6.