Problem 1 - Single Variable Calculus

Question

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+2} $$

Step-by-Step Solution

Answer

The given series converges because it meets all three conditions of the Alternating Series Test: the terms alternate in sign, the absolute value of the terms decreases, and the limit of the absolute value of the terms as n approaches infinity is 0.

Step 1: Verify that the terms alternate in sign

The given series is: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+2} \] The term \((-1)^{n-1}\) is responsible for alternating the sign of the terms. This ensures that the terms alternate in sign for every value of n.

Step 2: Check that the absolute value of the terms decreases

Now, we will check if the absolute value of the terms decreases. The absolute value of the terms is: \[ | b_n | = \left| \frac{(-1)^{n-1}}{n+2} \right| = \frac{1}{n+2} \] We need to check if \(| b_{n+1} | \le | b_n | \): \[ \frac{1}{(n+1)+2} \le \frac{1}{n+2} \] Simplifying, we get: \[ \frac{1}{n+3} \le \frac{1}{n+2} \] Since \(n+3 > n+2\), for all n, we can say that the absolute value of the terms is decreasing.

Step 3: Calculate the limit of the absolute value of the terms as n approaches infinity

We need to find the limit of the absolute value of the terms as n approaches infinity: \[ \lim_{n \to \infty} | b_n | = \lim_{n \to \infty} \frac{1}{n+2} \] As n approaches infinity, the term \(n+2\) will also approach infinity, which implies that the fraction will approach 0: \[ \lim_{n \to \infty} \frac{1}{n+2} = 0 \] Since all three conditions of the Alternating Series Test are met, we can conclude that the given series converges.

Key Concepts

Convergence Tests

Understanding convergence tests is essential when dealing with infinite series in calculus. These tests help determine if a given series converges or diverges. Convergence means that as more terms of the series are added, they get closer and closer to a specific value, rather than growing indefinitely or oscillating without settling. Different tests are applied depending on the properties of the series:

• Direct Comparison Test: Compare the series to a known convergent or divergent series.
• Limit Comparison Test: Take the limit of the ratio of the terms of the series.
• Ratio Test: Examine the limit of the ratio of consecutive terms.
• Alternating Series Test: Used specifically when the series terms alternate in sign.

Each test has specific conditions, and applying the correct test is crucial to finding the right solution.

Series Convergence

Series convergence refers to the behavior of a series as the number of terms increases without bound. For a series to converge, it must approach a particular sum. To understand convergence, picture a series as an infinite sum: \[S = a_1 + a_2 + a_3 + \cdots\]If this series reaches a finite number, it converges. If not, it diverges.

• Convergent series stabilize around a specific value.
• Divergent series either grow infinitely or fail to settle on a single value.
To test convergence, various methods are used based on the series' properties. Convergence is a key concept in calculus, ensuring that calculations involving infinite sums are meaningful.

Calculus Series

In calculus, series are used to represent and calculate functions and other mathematical quantities in terms of infinite sums of terms. Calculus series can take many forms, including:

• Geometric series: Involves terms in a constant ratio.
• Arithmetic series: Defined by a constant difference between terms.
• Harmonic series: Includes reciprocals of natural numbers.
• Alternating series: Term signs alternate between positive and negative.

Each type has unique properties and corresponds to specific convergence tests. Understanding how series work and using calculus tools allows us to approximate functions, solve complex problems, and explore infinite behaviors.

Alternating Series Test

The Alternating Series Test is a specific tool for examining the convergence of series whose terms alternate in sign. An alternating series looks like this:\[\sum (-1)^{n-1} b_n \]where each term's sign alternates. For an alternating series to converge, it must satisfy these conditions:

• The absolute values of the terms must continuously decrease.
• The limit of the absolute values as the term number goes to infinity must be zero.

If both criteria are met, the series converges.The given series, \[\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+2}\]satisfies these criteria because its terms decrease, and the limit of the terms as \(n \to \infty\) is zero. Therefore, using the alternating series test, the series converges.