Problem 1 - Calculus: Early Transcendentals

Question

(a) What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after \( n \) terms?

Step-by-Step Solution

Answer

An alternating series has terms that alternate in sign and converges if the terms decrease to zero. The remainder after \( n \) terms is at most the size of the next term.

Step 1: Understanding Alternating Series

An alternating series is a series whose terms alternate in sign. It can be generally expressed in the form \( \sum_{n=1}^{\infty} (-1)^{n-1} a_n \) or \( \sum_{n=1}^{\infty} (-1)^{n} a_n \) where \( a_n \) are positive, real numbers.

Step 2: Conditions for Convergence

An alternating series \( \sum_{n=1}^{\infty} (-1)^{n-1} a_n \) converges if two conditions are met: 1. The sequence \( a_n \) is monotonically decreasing, i.e., \( a_{n+1} \leq a_n \) for all \( n \) sufficiently large.2. The limit of \( a_n \) as \( n \) approaches infinity is zero, i.e., \( \lim_{n \to \infty} a_n = 0 \). These are known as the Alternating Series Test conditions.

Step 3: Understanding the Remainder Estimate

When a converging alternating series satisfies these conditions, the error or remainder \( R_n \) involved in truncating after \( n \) terms is less than or equal to the first omitted term. Mathematically, \( |R_n| \leq a_{n+1} \). This provides a useful bound for estimating the accuracy of the series sum after \( n \) terms.

Key Concepts

Convergence Conditions

In mathematics, understanding when a series converges is crucial for analyzing its behavior and determining if it sums to a finite number. For alternating series, convergence conditions are quite specific. These series flip in sign between each consecutive term, which can be generally written as \( \sum_{n=1}^{\infty} (-1)^{n-1} a_n \) or \( \sum_{n=1}^{\infty} (-1)^{n} a_n \). An alternating series converges if the following two conditions are satisfied:

• The sequence \( a_n \) is monotonically decreasing, meaning that each term is less than or equal to the previous term when \( n \) is sufficiently large. So, for a convergent alternating series, \( a_{n+1} \leq a_n \) holds for all large \( n \).
• The limit of \( a_n \) as \( n \) approaches infinity is zero. Formally, this is \( \lim_{n \to \infty} a_n = 0 \).

These two conditions are known as the Alternating Series Test, which, when satisfied, ensures the series converges. This test stands out because not all series converge even if their terms approach zero, making these conditions specifically important for alternating series.

Alternating Series Test

The Alternating Series Test is a powerful tool for determining whether an alternating series converges. It provides a clear criterion to check before concluding the behavior of such series. As discussed, an alternating series takes the form \( \sum_{n=1}^{\infty} (-1)^{n-1} a_n \) or \( \sum_{n=1}^{\infty} (-1)^{n} a_n \), where each term swings between positive and negative values.To use the Alternating Series Test effectively, consider these steps:

• First, verify that the sequence \( a_n \) is decreasing. This typically involves showing \( a_{n+1} \leq a_n \) from some point onward. Carefully examine if, after a specific term, each subsequent term is smaller than the last.
• Next, ensure that the limit of the terms \( a_n \) approaches zero at infinity. Calculate \( \lim_{n \to \infty} a_n \) and confirm that this result is zero.

If both criteria are satisfied, the series converges, thanks to the alternating components balancing the sum's growth and decline, ultimately steering it to converge to a finite point.

Remainder Estimate

The Remainder Estimate is a vital part of understanding the accuracy of sums from an alternating series truncated after a specific number of terms. If a series converges by the Alternating Series Test, we can gain insight into how the sum approximates the true solution when we only stop after a limited number of terms \( n \). The remainder, or error, \( R_n \), is what remains to reach the exact infinite sum.The remarkable feature of convergent alternating series is the simple bound rule:

• The absolute value of the remainder \( R_n \) is less than or equal to the first omitted term, \( a_{n+1} \). This tells us: \( |R_n| \leq a_{n+1} \).

In practical terms, this estimate allows us to say that if we stop at the \( n \)-th term, the maximum error we've committed is at most the next term in the series. This is a powerful and reassuring property, giving us confidence in approximate calculations when the full series sum is infeasible to compute.