Chapter 7: Q 47. (page 615)

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

Short Answer

Expert verified

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 $|r| < 1$ .

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 $\frac{3}{1 - \frac{1}{2}}$ , that is 6.

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Determine whether the series $\sum_{k = 0}^{\infty} \frac{3}{2^{k}}$ 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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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.

One App. One Place for Learning.

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