Chapter 7: Q 52. (page 615)

Determine whether the series $\sum_{k = 0}^{\infty} e {(\frac{π}{3})}^{k}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series $\sum_{k = 0}^{\infty} e {(\frac{π}{3})}^{k}$ diverges.

Step by step solution

Step 1. Given information.

Given a series $\sum_{k = 0}^{\infty} e {(\frac{π}{3})}^{k}$ .

Step 2. Find if the series converges or not.

The series $\sum_{k = 0}^{\infty} e {(\frac{π}{3})}^{k}$ is in the standard form $\sum_{k = 0}^{\infty} c r^{k}$ for a geometric series with $c = e$ and $r = \frac{π}{3}$ .

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

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

Therefore, the series $\sum_{k = 0}^{\infty} e {(\frac{π}{3})}^{k}$ diverges.

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Determine whether the series $\sum_{k = 0}^{\infty} e {(\frac{π}{3})}^{k}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the series converges or not.

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Step by step solution

Step 1. Given information.

Step 2. Find if the series converges or not.

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Most popular questions from this chapter

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Determine whether the series $\sum_{k = 0}^{\infty} e {(\frac{π}{3})}^{k}$ converges or diverges. Give the sum of the convergent series.