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.

Accepted Answer

Recall the Alternating Series Remainder Theorem, which states that for an alternating series with terms $$ a_n $$ that decrease in magnitude and tend to zero, the remainder after $$ n $$ terms, $$ R_n = S - S_n $$, satisfies $$ |R_n| \leq |a_{n+1}| . $$Understand that the partial sums $$ S_n = a_1 - a_2 + a_3 - \cdots + (-1)^{n+1} a_n $$ alternate around the actual sum $$ S $$, meaning each successive partial sum overestimates or underestimates $$ S  by an $$amount less than the magnitude of the next term. Since the terms $$ |a_n| $$ are nonincreasing, the difference between consecutive partial sums $$ |S_{n+1} - S_n| = |a_{n+1}| $$ provides an upper bound on how much the sum can change, and thus bounds the remainder. Because the series alternates in sign, the error $$ R_n $$ does not exceed the size of the next term in magnitude; the partial sums 'zig-zag' closer to the true sum, never overshooting by more than $$ |a_{n+1}| . $$Therefore, the magnitude of the remainder after $$ n $$ terms is at most the magnitude of the first neglected term, which ensures $$ |R_n| \leq |a_{n+1}| .$$

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.

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Key Concepts

Alternating Series

Nonincreasing Magnitude of Terms

Remainder Estimation in Alternating Series