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Chapter 7: Q. 2 (page 655)

Fill in the blanks.
For r_ , the sequence $\{r^{k}\}$ converges to _.

Short Answer

Expert verified

The required answer is for $r = 1$ , the sequence $\{r^{k}\}$ converges to 1 and for $|r| < 1$ , the sequence converges to 0.

Step by step solution

01

Step 1. Given Information

The given data is that geometric sequence converges.

02

Step 2. Explanation

Using the concept of convergence and divergence of Geometric series.

Let ${\{r^{k}\}}_{k = 0}^{\infty}$ be a geometric sequence. Then for $r = 1$ , the sequence converges to 1 and for $|r| < 1$ , the sequence converges to 0.

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Most popular questions from this chapter

Which p-series converge and which diverge?

An Improper Integral and Infinite Series: Sketch the function $f (x) = \frac{1}{x}$ for x ≥ 1 together with the graph of the terms of the series $\sum_{k = 1}^{\infty} \frac{1}{k} .$ Argue that for every term $S_{n}$ of the sequence of partial sums for this series, $S_{n} > \int_{1}^{n + 1} \frac{1}{x} d x$ . What does this result tell you about the convergence of the series?

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

Given a series $\sum_{k = 1}^{\infty} a_{k}$ , in general the divergence test is inconclusive when $a_{k} \to 0$ . For a geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

Explain how you could adapt the integral test to analyze a series $\sum_{k = 1}^{\infty} f (k)$ in which the function $f : [1, \infty) \to ℝ$ is continuous, negative, and increasing.

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