Problem 35

Question

Which of the series in Exercises $15 - 48$ converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } \frac { \cos n \pi } { n \sqrt { n } } $$

Step-by-Step Solution

Verified

Answer

The series converges absolutely because it is an alternating series with terms $ \left| a_n \right| = \frac{1}{n^{3/2}} $, which is convergent.

1Step 1: Analyze the General Term

The general term of the series is given by $ a_n = \frac{\cos(n\pi)}{n\sqrt{n}} $. Note that $ \cos(n\pi) $ alternates between $ -1 $ and $ 1 $ depending on whether $ n $ is even or odd. Hence, the series is alternating.

2Step 2: Check Absolute Convergence

We check absolute convergence by considering $ \left| a_n \right| = \frac{1}{n^{3/2}} $. The series $ \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} $ is a p-series with $ p = 3/2 > 1 $. Therefore, $ \sum_{n=1}^{\infty} \left| a_n \right| $ converges.

3Step 3: Conclusion on Absolute Convergence

Since the series $ \sum_{n=1}^{\infty} \left| a_n \right| $ converges, the original series $ \sum_{n=1}^{\infty} a_n $ converges absolutely.

Key Concepts

Alternating SeriesP-SeriesConvergence TestsSeries Analysis

Alternating Series

An alternating series is a series whose terms alternate in sign. In other words, the sequence of terms goes back and forth between positive and negative values. This can happen when the terms of the series are governed by a factor such as $ \cos(n\pi) $ which equals $-1$ or $1$ depending on whether $n$ is odd or even, respectively. Alternating series are important because they can sometimes converge even when the corresponding series of absolute values does not.

To analyze an alternating series, one must consider both convergence (whether it approaches a finite limit) and absolute convergence (whether the sum of absolute values converges).
For example, the Leibniz Test is a common method to test the convergence of alternating series.

Altering series are frequently studied due to these interesting properties, which help understand deeper convergence behaviors of series.

P-Series

A p-series is a specific type of mathematical series characterized by \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]where $ p $ is a positive constant. The behavior of a p-series depends heavily on the value of $ p $.

If $ p > 1 $, the p-series converges.
If $ p \leq 1 $, the series diverges.

This series is widely used as a benchmark for analyzing other series due to its simple form and well-defined convergence criteria. For example, with $ p=3/2 $, as in our exercise, the series converges. This allows us to make conclusions about the convergence of more complex series when they can be compared to a p-series.

Convergence Tests

Convergence tests are techniques used to determine whether a series converges. There are multiple tests available, each suited for different types of series. Some of the common ones are:

Alternating Series Test (Leibniz Test): This checks the convergence of alternating series based on the size of terms and whether they decrease to zero.
Ratio Test: Frequently used to assess series with factorials or exponential functions.
Root Test: Useful for series involving roots or exponential growth.
P-Series Test: Specifically for p-series, based on the power of the denominator.

For each series, picking the correct test depends on the form and characteristics of the series' general term. For example, in our exercise, the p-series test confirmed convergence by comparing to a known convergent p-series.

Series Analysis

Series analysis involves a thorough evaluation of series to determine their convergence, divergence, or absolute convergence properties. This can include identifying and applying suitable convergence tests, recognizing underlying patterns, or decomposing a series into simpler components for analysis.
Key steps in series analysis might include:

Identifying whether the series is alternating or not.
Checking for absolute convergence by examining a related p-series or using integral tests.
Concluding based on the results of convergence tests, making sure to justify the use of each method employed.

By systematically applying these techniques, one gains a comprehensive understanding of a series' behavior.

Problem 35

Other exercises in this chapter

Problem 35

In Exercises $1-36,$ (a) find the series' radius and interval of convergence. For what values of $x$ does the series converge (b) absolutely (c) conditional

View solution

Problem 35

In Exercises $17-46,$ use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty} \frac{(n+3) !}{3 !

View solution

Problem 35

In Exercises $31-38,$ use the $n$ th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=1}^{\i

View solution

Problem 35

Which of the sequences $\left\\{a_{n}\right\\}$ converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1-5 n^{4}}{n^{4}+8 n^{

View solution