Problem 55 - Thomas Calculus

Question

Which series in Exercises \(53-76\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3}{2^{n}}$$

Step-by-Step Solution

Answer

The series converges, and its sum is 1.5.

Step 1: Identify the Series Type

We are given the series \(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3}{2^{n}}\). Notice that this is an alternating series due to the \((-1)^{n+1}\) term which alternates signs of each term.

Step 2: Apply Alternating Series Test

The Alternating Series Test states that an alternating series \(\sum (-1)^{n} a_n\) converges if the sequence \(a_n\) is positive, decreasing, and approaches zero as \(n\) approaches infinity. Here, \(a_n = \frac{3}{2^n}\), which is positive and decreases as \(n\) increases.

Step 3: Verify Terms Approach Zero

Check \(\lim_{n \to \infty} \frac{3}{2^n} = 0\). As \(n\) goes to infinity, \(2^n\) becomes very large making \(\frac{3}{2^n}\) get closer to 0. All conditions of the test are satisfied, so the series converges.

Step 4: Determine the Sum of the Convergent Series

Since the series \(\sum_{n=1}^{\infty} \frac{3}{2^n}\) is a geometric series with the first term \(a = \frac{3}{2}\) and common ratio \(r = \frac{1}{2}\), its sum is computed using the formula for the sum of an infinite geometric series: \(S = \frac{a}{1-r}\).

Step 5: Compute the Sum

Substitute the values into the formula, \(S = \frac{\frac{3}{2}}{1-\frac{1}{2}} = \frac{\frac{3}{2}}{\frac{1}{2}} = 3\.0\). However, the alternating nature of the original series affects the sum, so substitute into the sum of an alternating series.

Step 6: Adjust for Alternating Sign

The sum of the alternating series can be found by taking half of the sum of the absolute value of the geometric series: \(S = 3 / 2 = 1.5\).

Key Concepts

Convergence

Convergence in a series is when the sum of an infinite series approaches a finite value as the number of terms goes to infinity. In the given alternating series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{3}{2^{n}} \), convergence is determined using the **Alternating Series Test**.

• This test looks at each term's nature: terms must be positive and decreasing.
• Also, the terms should approach zero.

Our sequence, \( a_n = \frac{3}{2^n} \), fulfills these criteria - it's positive and decreases as \( n \) increases.
The limit \( \lim_{n \to \infty} \frac{3}{2^n} = 0 \) confirms the terms approach zero, resulting in convergence for the series.

Divergence

Divergence is when the sum of an infinite series does not approach a finite number, instead, it either grows without bound or oscillates indefinitely. For a series to diverge, it often fails the conditions required for convergence:

• If terms do not approach zero as the number \( n \) increases, the series diverges.
• For an alternating series like ours, failing the positive, decreasing behavior or \( a_n \rightarrow 0 \) criterion means divergence.

In this exercise, our series does not diverge because it meets all the convergence conditions.

Geometric Series

A geometric series is where each term is a fixed multiple (or ratio) of the previous term. The series \( \sum_{n=1}^{\infty} \frac{3}{2^n} \) is identified as geometric without the alternating sign:

• First term: \( a = \frac{3}{2} \)
• Common ratio: \( r = \frac{1}{2} \)

For an infinite geometric series, the sum can be calculated if \( |r| < 1 \). This formula is \( S = \frac{a}{1-r} \).
The series \( \sum \frac{3}{2^n} \) would have an infinite sum of 3 without alternating signs.
Yet, the original series in the exercise includes sign alteration, affecting the sum.

Infinite Series

An infinite series is a sum of unlimited terms written as \( \sum_{n=1}^{\infty} a_n \). Often people find infinite series challenging since they involve adding an endless list of numbers.
The key to dealing with them is understanding **convergence** and **divergence** conditions.

• Convergence implies the sum approaches a specific number.
• Divergence means no finite sum exists.

The exercise deals with an alternating infinite series, assessed using specific tests to determine if the sum is finite.
Understanding these properties helps in evaluating the behavior of diverse infinite series beyond this specific problem.