Q6P Test the following series for co... [FREE SOLUTION]

Chapter 1: Q6P (page 17)

Test the following series for convergence
${鈭�}_{n - 1}^{鈭�} \frac{{(- 1)}^{n} n}{n + 5}$

Short Answer

Expert verified

The alternating series ${鈭�}_{n - 1}^{鈭�} \frac{{(- 1)}^{n} n}{n + 5}$ diverges.

Step by step solution

Significance of alternating series

If the terms' absolute values gradually decline to zero, or if $| a_{n + 1} | 鈮� | a_{n} |$ and $\underset{n 鈫� 鈭�}{l i m} a_{n} = 0$ , then an alternating series converges.

Use a test to check for convergence

Consider the alternating series,

${鈭�}_{n - 1}^{鈭�} \frac{{(- 1)}^{n} n}{n + 5}$

Use the below test to check the series for convergence:

$| a_{n + 1} | 鈮� | a_{n} |$ and $\underset{n 鈫� 鈭�}{l i m} a_{n} = 0 .$

The series converges if the condition is met.

$a_{n} = \frac{n {(- 1)}^{n}}{n + 5} |a_{n}| = \frac{n}{n + 5} |a_{n + 1}| = \frac{n + 1}{n + 6}$

Perform the test to check if both conditions are met

Let check if $|a_{n}| > |a_{n + 1}|$ and using the following method.

For $|a_{n + 1}| - |a_{n}| < 0$ this will mean that $|a_{n}| > |a_{n + 1}|$

For $|a_{n + 1}| - |a_{n}| > 0$ this will mean that $|a_{n}| < |a_{n + 1}|$

$|a_{n + 1}| - |a_{n}| = \frac{n + 1}{n + 6} - \frac{n}{n + 5} = \frac{n^{2} + 6 n + 5 - n^{2} - 6 n}{(n + 5) (n + 6)} = \frac{5}{(n + 5) (n + 6)}$

Now, for $n 鈮� 1, |a_{n + 1}| - |a_{n}| > 0, |a_{n}| < |a_{n + 1}|$

Therefore, we conclude that $|a_{n + 1}| > |a_{n}|$ for $n 鈮� 1$

As the first condition is not met, so the series diverges.

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91影视

Test the following series for convergence
${鈭�}_{n - 1}^{鈭�} \frac{{(- 1)}^{n} n}{n + 5}$

Short Answer

Step by step solution

Significance of alternating series

Use a test to check for convergence

Perform the test to check if both conditions are met

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Test the following series for convergence鈭�n-1鈭�(-1)nnn+5

Short Answer

Step by step solution

Significance of alternating series

Use a test to check for convergence

Perform the test to check if both conditions are met

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Medical Physics

Conservation of Energy and Momentum

Wave Optics

Atoms and Radioactivity

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

Test the following series for convergence
${鈭�}_{n - 1}^{鈭�} \frac{{(- 1)}^{n} n}{n + 5}$