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.


59. ∑ (k = –3 to ∞) 4 / ((4k – 3)(4k + 1))

Verified step by step guidance

1. Start by expressing the general term of the series: \( a_k = \frac{4}{(4k - 3)(4k + 1)} \). Our goal is to rewrite this term in a form that reveals telescoping behavior.
2. Use partial fraction decomposition to break \( a_k \) into simpler fractions. Assume \( \frac{4}{(4k - 3)(4k + 1)} = \frac{A}{4k - 3} + \frac{B}{4k + 1} \) and solve for constants \( A \) and \( B \).
3. Once you find \( A \) and \( B \), rewrite \( a_k \) as the difference of two fractions, which will help terms cancel out when summing from \( k = -3 \) to \( n \).
4. Write the partial sum \( S_n = \sum_{k=-3}^n a_k \) using the decomposed form. Observe how most terms cancel out due to the telescoping nature, leaving only a few terms from the start and end of the sum.
5. Express \( S_n \) explicitly in terms of \( n \), then analyze \( \lim_{n \to \infty} S_n \) to determine whether the series converges or diverges.

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 Sums and Their Formulas

The partial sum Sₙ of a series is the sum of its first n terms. Finding a closed-form expression for Sₙ helps in evaluating the behavior of the series as n approaches infinity, which is essential for determining convergence or divergence.

Limit of a Sequence

The limit of the sequence of partial sums, limₙ→∞ Sₙ, determines the sum of an infinite series if it exists. If this limit is finite, the series converges; otherwise, it diverges. Understanding limits is crucial for evaluating infinite series.