Question

Determine whether the series $\sum _{k=0}^{\infty }e{\left(\frac{\mathrm{\pi }}{3}\right)}^k$converges or diverges. Give the sum of the convergent series.

Short answer

The series $\sum _{k=0}^{\infty }e{\left(\frac{\mathrm{\pi }}{3}\right)}^k$diverges.

Step-by-step solution

Step 1. Given information.

Given a series $\sum _{k=0}^{\infty }e{\left(\frac{\mathrm{\pi }}{3}\right)}^k$.

Step 2. Find if the series converges or not.

The series $\sum _{k=0}^{\infty }e{\left(\frac{\mathrm{\pi }}{3}\right)}^k$is in the standard form $\sum _{k=0}^{\infty }c{r}^k$for a geometric series with $c=e$and $r=\frac{\mathrm{\pi }}{3}$.

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

Note that localid="1648959292348" $\mathrm{\pi }\approx 3.14$. It follows that $\frac{\mathrm{\pi }}{3}>1$.

Therefore, the series $\sum _{k=0}^{\infty }e{\left(\frac{\mathrm{\pi }}{3}\right)}^k$diverges.