Which Of The Following Series Is Absolutely Convergent

Which of the Following Series is Absolutely Convergent? A Comprehensive Guide

Determining the absolute convergence of an infinite series is a crucial concept in calculus and real analysis. It dictates whether a series converges not just to a value, but does so in a particularly strong sense, unaffected by rearrangements of its terms. This article will delve deep into the methods used to determine absolute convergence, exploring various tests and providing detailed examples to solidify understanding. We'll tackle the question directly and systematically examine how to identify absolutely convergent series among a given set.

Understanding Absolute Convergence

Before we dive into specific series, let's establish a firm grasp of what absolute convergence means. A series $\sum_{n=1}^{\infty} a_n$ is said to be absolutely convergent if the series of the absolute values of its terms, $\sum_{n=1}^{\infty} |a_n|$, converges. Crucially, if a series is absolutely convergent, it is also convergent. However, the converse is not always true; a series can be convergent but not absolutely convergent (this is known as conditional convergence).

Think of it like this: absolute convergence is a stronger condition than regular convergence. If a series converges absolutely, it's robust—its sum remains unchanged even if you shuffle its terms. A conditionally convergent series, on the other hand, is more delicate; changing the order of its terms can change its sum, or even cause it to diverge.

Key Tests for Absolute Convergence

Several tests can help determine whether a series converges absolutely. We'll explore some of the most important:

1. The Comparison Test

The comparison test is a fundamental tool for determining convergence. If we have two series, $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$, where $0 \le a_n \le b_n$ for all $n$, then:

• If $\sum_{n=1}^{\infty} b_n$ converges, then $\sum_{n=1}^{\infty} a_n$ also converges.
• If $\sum_{n=1}^{\infty} a_n$ diverges, then $\sum_{n=1}^{\infty} b_n$ also diverges.

This test is particularly useful for determining absolute convergence. If we can find a convergent series $\sum_{n=1}^{\infty} b_n$ such that $|a_n| \le b_n$ for all $n$, then $\sum_{n=1}^{\infty} |a_n|$ converges, implying that $\sum_{n=1}^{\infty} a_n$ is absolutely convergent.

2. The Limit Comparison Test

A refinement of the comparison test, the limit comparison test is often easier to apply. Given two series $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ with positive terms, if the limit

$\lim_{n\to\infty} \frac{a_n}{b_n} = L$

exists and $0 < L < \infty$, then both series converge or both diverge together. Again, this can be applied to $|a_n|$ to determine absolute convergence.

3. The Ratio Test

The ratio test is a powerful tool for series with terms involving factorials or exponentials. For a series $\sum_{n=1}^{\infty} a_n$, we consider the limit:

$\lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = L$

• If $L < 1$, the series converges absolutely.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

4. The Root Test

Similar to the ratio test, the root test examines the limit:

$\lim_{n\to\infty} \sqrt[n]{|a_n|} = L$

• If $L < 1$, the series converges absolutely.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

5. The Integral Test

The integral test connects the convergence of a series to the convergence of an improper integral. If $f(x)$ is a positive, continuous, and decreasing function on $[1, \infty)$ such that $f(n) = a_n$, then $\sum_{n=1}^{\infty} a_n$ converges if and only if $\int_{1}^{\infty} f(x) , dx$ converges. This test can be used to assess absolute convergence by applying it to $|a_n|$.

Analyzing Specific Series for Absolute Convergence

Let's consider some examples to illustrate the application of these tests. Suppose we are given several series and asked to determine which are absolutely convergent. We will systematically analyze each one.

Example 1: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$

This series is an alternating series. Let's consider the absolute values of its terms: $\sum_{n=1}^{\infty} \frac{1}{n^2}$. This is a p-series with p=2, which is known to converge. Therefore, $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ is absolutely convergent.

Example 2: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$

Again, this is an alternating series. Let's check for absolute convergence: $\sum_{n=1}^{\infty} \frac{1}{n}$. This is the harmonic series, which is known to diverge. However, the alternating harmonic series converges (by the alternating series test). Therefore, $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ is conditionally convergent (convergent but not absolutely convergent).

Example 3: $\sum_{n=1}^{\infty} \frac{n^2}{2^n}$

We can apply the ratio test here:

$\lim_{n\to\infty} \left| \frac{(n+1)^2}{2^{n+1}} \cdot \frac{2^n}{n^2} \right| = \lim_{n\to\infty} \left| \frac{(n+1)^2}{2n^2} \right| = \frac{1}{2} < 1$

Since the limit is less than 1, the series converges absolutely.

Example 4: $\sum_{n=1}^{\infty} \frac{(-1)^n n!}{n^n}$

Applying the ratio test:

$\lim_{n\to\infty} \left| \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} \right| = \lim_{n\to\infty} \left| \frac{(n+1)n^n}{(n+1)^{n+1}} \right| = \lim_{n\to\infty} \left( \frac{n}{n+1} \right)^n \frac{1}{n+1} = \frac{1}{e} < 1$

Again, the limit is less than 1, thus the series is absolutely convergent.

Example 5: $\sum_{n=1}^{\infty} \frac{\cos(n\pi)}{n}$

This series can be rewritten as $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$, which is the alternating harmonic series, as discussed above. It is conditionally convergent.

Conclusion

Determining absolute convergence is a critical skill in understanding infinite series. By mastering the various tests—comparison, limit comparison, ratio, root, and integral—and applying them systematically, we can confidently analyze the convergence behavior of a wide range of series. Remember, absolute convergence implies convergence, but the converse is not always true. The examples presented illustrate how to apply these tests and differentiate between absolute and conditional convergence. This comprehensive guide should equip you to tackle a variety of problems related to absolute convergence of infinite series. Further exploration of advanced techniques and special functions can extend this understanding even further. Remember to always consider the properties of the series and select the most appropriate convergence test. Practice is key to mastering these concepts and becoming proficient in determining absolute convergence.