Q 47. - Calculus

Question

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

Step-by-Step Solution

Answer

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

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 $\sum _{k=0}^{\infty }c{r}^k$ for a geometric series with $c=3$ and $r=\frac{1}{2}$.

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

Since $r=\frac{1}{2}$, 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.