Problem 57 - 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. $$\sum_{n=0}^{\infty} \cos \left(\frac{n \pi}{2}\right)$$

Step-by-Step Solution

Answer

The series converges to 0 as it alternates in a pattern yielding a zero sum.

Step 1: Analyze the General Term

We need to analyze the general term of the series given by \( a_n = \cos\left(\frac{n \pi}{2}\right) \). Let us compute the initial terms to discern any pattern:- For \( n = 0 \), \( a_0 = \cos(0) = 1 \).- For \( n = 1 \), \( a_1 = \cos\left(\frac{\pi}{2}\right) = 0 \).- For \( n = 2 \), \( a_2 = \cos(\pi) = -1 \).- For \( n = 3 \), \( a_3 = \cos\left(\frac{3\pi}{2}\right) = 0 \). Continue this process to confirm the pattern: \( a_n = (1, 0, -1, 0) \). Thus, the series alternates.

Step 2: Identify the Series Pattern

The terms follow a repeating pattern of \( (1, 0, -1, 0) \). Notice, for increments of 4 in \( n \), the sum through this cycle is:\[ 1 + 0 + (-1) + 0 = 0 \].Understanding this shows that every four terms sum to zero.

Step 3: Determine the Behavior of the Series

Since each complete cycle of four terms sums to zero and this pattern repeats indefinitely without affecting the sum, the series converges.

Step 4: Evaluate the Sum of the Series

Given the repeating nature with zero sum for each cycle, we can conclude that:\[ \sum_{n=0}^{\infty} \cos\left(\frac{n \pi}{2}\right) = 0 \].As any prefix sum of a set of complete cycles will remain zero, therefore, the sum of the series is 0.

Key Concepts

Alternating Series

Alternating series have terms that switch signs, typically creating a zigzag pattern. These series often alternate between positive and negative values. For instance, they might start with a positive term, followed by a negative, then positive, and so on. This type of series can potentially converge or diverge, depending on additional properties the series may have.
To determine convergence of an alternating series, the Alternating Series Test (or Leibniz Test) is commonly applied. This test states an alternating series converges if:

• The magnitude of the terms \( a_n \) decreases to 0 as \( n \) approaches infinity.
• The limit of each term as \( n \) goes to infinity is zero.

In the analyzed series, the terms \( a_n = \cos\left(\frac{n\pi}{2}\right) \) exhibit an alternating pattern, but we must analyze further behaviors to conclude its convergence.

General Term Analysis

The general term analysis is crucial for understanding any series. Here, we examine the expression for the n-th term, \( a_n = \cos\left(\frac{n\pi}{2}\right) \), to figure out the behavior of the sequence. By calculating the initial terms, you can identify a pattern:

• For \( n = 0 \), \( a_0 = \cos(0) = 1 \).
• For \( n = 1 \), \( a_1 = \cos\left(\frac{\pi}{2}\right) = 0 \).
• For \( n = 2 \), \( a_2 = \cos(\pi) = -1 \).
• For \( n = 3 \), \( a_3 = \cos\left(\frac{3\pi}{2}\right) = 0 \).

This analysis reveals a repeating cycle: \( (1, 0, -1, 0) \). Understanding these repeating terms is essential in evaluating the convergence or divergence of the series.

Sum of Series

After establishing the behavior of the terms in the series, calculating the total sum is the next step. In the examined series, each cycle of four terms gives zero: \[ 1 + 0 + (-1) + 0 = 0 \]When summed up across the entire series, our repeating pattern results in these four-term cycles canceling out to zero.
Thus, the total sum of this infinite series is zero. This conclusion might seem a bit paradoxical at first. Still, understanding the idea that a perfectly aligned zero sum at each cycle results in the convergence to a finite total, makes perfect sense in this context.

Pattern Recognition

Recognizing a pattern in the terms is instrumental for series analysis. Such patterns simplify complex behaviors and help in determining the series' convergence or divergence. In this particular series, we identified a distinct repeating pattern - \( (1, 0, -1, 0) \).
By spotting this pattern, we can predict how the series behaves as \( n \) grows. Patterns like these can quicken the convergence test process and allow deeper insights into the series' structure. Instead of assessing each term individually over infinite iterations, recognizing repeating sequences simplifies the problem significantly. This holistic view of term behavior benefits both mathematical intuition and problem-solving efficiency.