Chapter 8: Q 6. (page 704)

Find the interval of convergence of the power series
$\sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (x - 4)^{k}$

Short Answer

Expert verified

The interval of convergence of the power series $\sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (x - 4)^{k}$ is $(3, 5)$

Step by step solution

Given information

The power series is $\sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (x - 4)^{k}$

The ratio test for absolute convergence will be used to determine the convergence interval.

Let, the first assume $b_{k} = \frac{k!}{k^{k}} (x - 4)^{k}$ , therefore $b_{k + 1} = \frac{(k + 1)!}{(k + 1)^{k + 1}} (x - 4)^{k + 1}$

$\lim_{k \to \infty} |\frac{b_{k + 1}}{b_{k}}| = \lim_{k \to \infty} |\frac{\frac{(k + 1)!}{(k + 1)^{k + 1}} (x - 4)^{k + 1}}{\frac{k!}{k^{k}} (x - 4)^{k}}| = \lim_{k \to \infty} \frac{(k + 1) k^{k}}{(k + 1)^{k + 1}} | x - 4 | = \lim_{k \to \infty} {(\frac{k}{k + 1})}^{k} | x - 4 |$

The limit is $| x - 4 |$

The ratio test for absolute convergence will be used to determine the convergence interval when $| x - 4 | < 1$ that is $- 1 < x - 4 < 1$

As a result, we may write $- 1 < x - 4$ and $x - 4 < 1$

Implies that

$x > 3 a n d x < 5$

$x \in (3, 5)$

Now, because the intervals are limited, we examine the series' behavior at the ends.

When $x = 3$

${\sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (x - 4)^{k}|}_{x = 5} = \sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (5 - 4)^{k} = \sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (1)^{k} = \sum_{k = 1}^{\infty} \frac{k!}{k^{k}}$

The constant multiple, which diverges, is the consequence.

The interval of convergence of the power series

The interval of convergence of the power series $\sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (x - 4)^{k}$ is $(3, 5)$

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Find the interval of convergence of the power series
$\sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (x - 4)^{k}$

Short Answer

Step by step solution

Given information

The ratio test for absolute convergence will be used to determine the convergence interval.

Now, because the intervals are limited, we examine the series' behavior at the ends.

The interval of convergence of the power series

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Find the interval of convergence of the power series ∑k=1∞k!kk(x-4)k

Short Answer

Step by step solution

Given information

The ratio test for absolute convergence will be used to determine the convergence interval.

Now, because the intervals are limited, we examine the series' behavior at the ends.

The interval of convergence of the power series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Decision Maths

Discrete Mathematics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find the interval of convergence of the power series
$\sum_{k = 1}^{\infty} \frac{k!}{k^{k}} (x - 4)^{k}$