Convergent Series
Find the sums of the series in Exercises 19–24.

∑ (from n = 2 to ∞) -2/[n(n+1)]

검증된 단계별 안내

Recognize that the series is given by \( \sum_{n=2}^{\infty} \frac{-2}{n(n+1)} \). This is a rational function that can be decomposed using partial fractions.

Express the general term \( \frac{-2}{n(n+1)} \) as a difference of two simpler fractions: find constants \( A \) and \( B \) such that \( \frac{-2}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1} \).

Solve for \( A \) and \( B \) by multiplying both sides by \( n(n+1) \) and equating coefficients, which will allow you to rewrite the series as a telescoping series.

Write out the first few terms of the series after partial fraction decomposition to observe the telescoping pattern, where many terms cancel out.

Use the telescoping property to find the sum of the infinite series by taking the limit as \( N \to \infty \) of the partial sums \( S_N \), which simplifies to a finite value.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Convergent Series

A series is convergent if the sum of its infinite terms approaches a finite limit. Determining convergence often involves tests or recognizing known series forms. For this problem, understanding convergence ensures the sum exists and can be calculated.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique is useful for series like ∑ -2/[n(n+1)] because it allows rewriting terms to reveal telescoping behavior, simplifying the sum calculation.

Telescoping Series

A telescoping series is one where many terms cancel out when partial sums are expanded. Recognizing telescoping allows easy computation of the series sum by focusing on the first and last terms of the partial sums, avoiding complicated infinite sums.

Convergent SeriesFind the sums of the series in Exercises 19–24.∑ (from n = 2 to ∞) -2/[n(n+1)]