Chapter 1: Q6P (page 17)

Test the following series for convergence
$\sum_{n - 1}^{\infty} \frac{{(- 1)}^{n} n}{n + 5}$

Short Answer

Expert verified

The alternating series $\sum_{n - 1}^{\infty} \frac{{(- 1)}^{n} n}{n + 5}$ diverges.

Step by step solution

Significance of alternating series

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

Use a test to check for convergence

Consider the alternating series,

$\sum_{n - 1}^{\infty} \frac{{(- 1)}^{n} n}{n + 5}$

Use the below test to check the series for convergence:

$| a_{n + 1} | \leq | a_{n} |$ and $\underset{n \to \infty}{l i m} a_{n} = 0 .$

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 $n \geq 1, |a_{n + 1}| - |a_{n}| > 0, |a_{n}| < |a_{n + 1}|$

Therefore, we conclude that $|a_{n + 1}| > |a_{n}|$ for $n \geq 1$

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

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Test the following series for convergence
$\sum_{n - 1}^{\infty} \frac{{(- 1)}^{n} n}{n + 5}$

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

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Test the following series for convergence∑n-1∞(-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

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Test the following series for convergence
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