Problem 5 - Calculus Early Transcendentals

Question

Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.

Step-by-Step Solution

Answer

Answer: The magnitude of the remainder in an alternating series with nonincreasing in magnitude terms is less than or equal to the magnitude of the first neglected term because of the Alternating Series Remainder Theorem. This theorem states that for any such series that converges and has decreasing term magnitudes, the difference (remainder) between the true sum and the sum of the first n terms is bounded by the magnitude of the (n+1)-th term, which is the first neglected term in the series. Since consecutive partial sums have smaller differences between them due to changing signs and decreasing magnitudes, the first neglected term represents an upper bound for the magnitude of the remainder.

Step 1: Understanding an alternating series

An alternating series is a series in which its terms alternate in sign, that is, a series of the form \(a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + \dots\), where each term is non-negative. In general, an alternating series can be written as \(\sum^\infty_{n=1} (-1)^{n-1} a_n\), where \(a_n \geq 0\) for all \(n\).

Step 2: Define alternating series with nonincreasing terms

An alternating series with nonincreasing terms is an alternating series in which the magnitudes of each term are nonincreasing, i.e., \(a_n \geq a_{n+1}\), for all \(n\). It means that the terms are getting smaller in magnitude with each new term.

Step 3: Recall the Alternating Series Remainder Theorem

The Alternating Series Remainder Theorem states that if an alternating series with nonincreasing term magnitudes meets the following two conditions: 1. \(a_n \geq a_{n+1}\), for all \(n\) 2. \(a_n \rightarrow 0\) as \(n \rightarrow \infty\) Then the series converges. Moreover, if \(R_n\) is the remainder after the \(n\)-th term, i.e., the difference between the true sum of the series and the sum of the first \(n\) terms, then the magnitude of the remainder is less than or equal to the magnitude of the first neglected term, i.e., \(|R_n| \leq a_{n+1}\).

Step 4: Explaining the relationship between the remainder and the first neglected term

Consider an alternating series that satisfies the conditions of the Alternating Series Remainder Theorem. Let the sum of the first \(n\) terms be \(S_n\), the true sum of the series be \(S\), and \(R_n = S - S_n\) be the remainder after the \(n\)-th term. Since each term in the series is getting smaller in magnitude and constantly changing in sign, adding or subtracting the next term in the series will create a new partial sum that is either closer to the true sum or, at least, not farther away from it than the previous partial sum. This implies that the difference between consecutive partial sums, which is the next term in the series, represents an upper bound of the difference (the remainder) between the sum of those first few terms and the true sum. Since, by definition, the first neglected term is equal to the \((n+1)\)-th term, we can write \(|R_n| \leq a_{n+1}\) , which shows that the magnitude of the remainder is less than or equal to the magnitude of the first neglected term.