Chapter 7: Q 51. (page 615)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

Step 2. Find if the series converges or not.

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

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

Note that $2 < e < 3$ . It follows that $\frac{e}{3} < 1$ .

Therefore, the series localid="1648958893904" $\sum_{k = 0}^{\infty} π {(\frac{e}{3})}^{k}$ 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 $\sum_{k = 0}^{\infty} π {(\frac{e}{3})}^{k}$ converges to $\frac{π}{1 - \frac{e}{3}}$ , that is $\frac{3 π}{3 - e}$ .

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Determine whether the series $\sum_{k = 0}^{\infty} π {(\frac{e}{3})}^{k}$ 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.

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

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Step by step solution

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

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Most popular questions from this chapter

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