Chapter 7: Q. 29 (page 639)

In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.
$\sum_{k = 0}^{\infty} \frac{5^{k}}{k!}$

Short Answer

Expert verified

The series converges.

Step by step solution

Step 1. Given information.

The given series is $\sum_{k = 0}^{\infty} \frac{5^{k}}{k!}$ .

Step 2. Ratio Test.

According to the given series,

$a_{k + 1} = \frac{5^{k + 1}}{(k + 1)!} \frac{a_{k + 1}}{a_{k}} = \frac{\frac{5^{k + 1}}{(k + 1)!}}{\frac{5^{k}}{k!}} = \frac{k! 5^{k + 1}}{5^{k} (k + 1)!} = \frac{5}{(k + 1)}$

Step 3. Take limits.

On Taking limits,

$\lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} = \lim_{k \to \infty} \frac{5}{(k + 1)} = 5 (\lim_{k \to \infty} \frac{1}{(k + 1)}) = 5 (0) = 0 Since, L < 1,$

Therefore, the series converges.

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In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.
$\sum_{k = 0}^{\infty} \frac{5^{k}}{k!}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Ratio Test.

Step 3. Take limits.

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In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.∑k=0∞5kk!

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Ratio Test.

Step 3. Take limits.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Logic and Functions

Probability and Statistics

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.
$\sum_{k = 0}^{\infty} \frac{5^{k}}{k!}$