Problem 16 Test the following series for co... [FREE SOLUTION]

Chapter 1: Problem 16

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=0}^{\infty} \frac{2+(-1)^{n}}{n^{2}+7}$$

Short Answer

Expert verified

The series converges by the Alternating Series Test.

Step by step solution

Preliminary Test - n-th Term Test

Check if the limit of the series terms as n approaches infinity is zero. Calculate the limit: \ $ \lim_\{n\to\infty} \frac{2+(-1)^n}{n^2+7} $. Since \frac{2+(-1)^n}{n^2+7} \ approximates to \frac{2}{n^2}\ for large n, \ \lim_\{n \to \infty\} \frac{2 + (-1)^n}{n^2 + 7} = 0\. The limit is zero, so proceed to additional tests.

Choosing the Test - Alternating Series Test

Given the form of $ b_n = \frac{2 + (-1)^n}{n^2 + 7} $ where the series alternates due to \ (-1)^n \, use the Alternating Series Test (Leibniz Test). Verify if \ b_n \ is decreasing and if \ \lim_\{n\to\infty} b_n = 0 \.

Check Monotonic Decreasing Condition

Evaluate if $ b_n = \frac{2 + (-1)^n}{n^2 + 7} $ is decreasing. For large \ n \, \ b_n \ approximates \ \frac {2}{n^2} \ which is decreasing. Hence, for sufficiently large \ n \, \ b_n \ is monotonically decreasing.

Alternating Series Test Result

Since $ \lim_\{n\to\infty} \frac{2+(-1)^n}{n^2+7} = 0 $ and the terms are decreasing for sufficiently large \, the series satisfies both conditions of the Alternating Series Test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

n-th Term Test

The n-th term test is a preliminary check for determining if a series converges or diverges. For a series $\sum a_n$ to converge, the limit of its terms as $n$ approaches infinity must be zero. This can be expressed mathematically as $\lim_\{n\to\infty\} a_n = 0$. If the limit is not zero, the series diverges. In our given example, $\frac{2 + (-1)^n}{n^2 + 7}$, the terms include $(-1)^n$, which makes the sequence alternate. We calculate the limit:
\[ \lim_\{n\to\infty\} \frac{2 + (-1)^n}{n^2 + 7} = \frac{2}{n^2} \approx 0 \]
Since the limit approaches zero, the n-th term test tells us we need to conduct further tests to confirm convergence.

Alternating Series Test

An alternating series is one in which the terms alternate between positive and negative. The general form is $\sum (-1)^{n} b_n$, where $(b_n)$ are positive terms. The Alternating Series Test (also known as the Leibniz Test) includes two conditions:
1. $ \lim_\{n\to\infty\} b_n = 0 $
2. $ b_n $ is monotonically decreasing.
If both conditions are met, the series converges. For our series, $ b_n = \frac{2 + (-1)^n}{n^2 + 7}$, we already established that
\[ \lim_\{n\to\infty\} \frac{2+(-1)^n}{n^2+7} = 0 \]
Now, we need to check if $ b_n $ is monotonically decreasing.

Monotonic Function

A function or sequence is monotonic if it is either entirely non-increasing or non-decreasing. In our context, we need to check if $(b_n)$ is decreasing. For large $ n $, $ b_n$ approximates $\frac{2}{n^2}$, which clearly decreases as \ n \ increases. Thus, $ b_n $ is monotonically decreasing.
To summarize:

The limit of $ b_n $ as \ n \ approaches infinity is zero.
$ b_n $ is a decreasing sequence for sufficiently large \ n \.

Therefore, by the Alternating Series Test, the given series converges.

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