Chapter 12: Q 61. (page 825)

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} \frac{1}{2} 3^{k - 1}$

Short Answer

Expert verified

The given infinite geometric series $\sum_{k = 1}^{\infty} \frac{1}{2} 3^{k - 1}$ is diverges.

Step by step solution

Step 1. Write the given information.

The given geometric series is:

\sum_{k = 1}^{\infty} \frac{1}{2} 3^{k - 1}

Determine the first term and common ratio.

The first term is $(a_{1}) = \frac{1}{2}$

The common ratio is the ratio of successive terms:

$r = 3$

Step 3. Determine whether the infinite geometric series is converges or diverges.

If $|r| < 1$ then the infinite geometric series converges.

As we can see that $|3| > 1$ , therefore the given series not converges.

The given infinite geometric series diverges.

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In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} \frac{1}{2} 3^{k - 1}$

Short Answer

Step by step solution

Step 1. Write the given information.

Determine the first term and common ratio.

Step 3. Determine whether the infinite geometric series is converges or diverges.

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In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.∑k=1∞123k-1

Short Answer

Step by step solution

Step 1. Write the given information.

Determine the first term and common ratio.

Step 3. Determine whether the infinite geometric series is converges or diverges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Applied Mathematics

Decision Maths

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} \frac{1}{2} 3^{k - 1}$