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.


65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))

Verified step by step guidance

1. Identify the general term of the series: \( a_k = \frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+3}} \). This is a telescoping series because each term involves the difference of two expressions with consecutive indices shifted by 2.
2. Write the nth partial sum \( S_n \) as the sum of the first n terms: \[ S_n = \sum_{k=1}^n \left( \frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+3}} \right) \].
3. Expand the sum to observe cancellation: \[ S_n = \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{4}} \right) + \left( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{5}} \right) + \cdots + \left( \frac{1}{\sqrt{n+1}} - \frac{1}{\sqrt{n+3}} \right) \].
4. Group the positive and negative terms to see which terms cancel out. Notice that many intermediate terms will cancel, leaving only a few terms from the beginning and the end of the sum.
5. Express \( S_n \) in a simplified form involving only the first few terms and the last few terms that do not cancel. Then, evaluate \( \lim_{n \to \infty} S_n \) by analyzing the behavior of the remaining terms as \( n \) approaches infinity to determine if the series converges or diverges.

Key Concepts

Telescoping Series

A telescoping series is a series whose partial sums simplify because many terms cancel out. Typically, each term can be written as the difference of two expressions, causing intermediate terms to vanish when summed. This simplification helps find a closed-form expression for the nth partial sum.

Partial Sums and Their Limits

The partial sum Sₙ of a series is the sum of its first n terms. Evaluating the limit of Sₙ as n approaches infinity determines whether the series converges or diverges. If the limit exists and is finite, the series converges to that value.

Manipulating Radical Expressions

Working with terms involving square roots often requires rationalizing or rewriting expressions to simplify differences. Recognizing patterns in radicals can help express terms in a form suitable for telescoping, making it easier to identify cancellations in the series.