54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))

Verified step by step guidance

1. Start by expressing the general term of the series: \( a_k = \frac{1}{(k+6)(k+7)} \). Our goal is to rewrite this term in a form that allows telescoping.
2. Use partial fraction decomposition to break \( a_k \) into simpler fractions. Set \( \frac{1}{(k+6)(k+7)} = \frac{A}{k+6} + \frac{B}{k+7} \) and solve for constants \( A \) and \( B \).
3. After finding \( A \) and \( B \), rewrite \( a_k \) as \( \frac{A}{k+6} + \frac{B}{k+7} \). This will help identify terms that cancel out when summing.
4. Write the partial sum \( S_n = \sum_{k=1}^n a_k \) by substituting the decomposed form of \( a_k \). Observe how most terms cancel out (telescope), leaving only a few terms from the beginning and end.
5. Express \( S_n \) explicitly in terms of \( n \), then evaluate the limit \( \lim_{n \to \infty} S_n \) to determine whether the series converges and find its sum if it does.

Key Concepts

Telescoping Series

A telescoping series is a series whose partial sums simplify by canceling intermediate terms, leaving only a few terms from the beginning and end. This simplification makes it easier to find a formula for the nth partial sum and analyze convergence.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions. For series like 1/((k+6)(k+7)), this technique helps rewrite terms so that the series telescopes, enabling easier summation and limit evaluation.

Limit of Partial Sums and Convergence

The sum of an infinite series is the limit of its partial sums as n approaches infinity. If this limit exists and is finite, the series converges; otherwise, it diverges. Evaluating this limit determines the series' value or divergence.