67–70. Formulas for sequences of partial sums Consider the following infinite series.


c.Make a conjecture for the value of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Verified step by step guidance

1. Start by examining the general term of the series: \(\frac{2}{(2k - 1)(2k + 1)}\). The goal is to simplify this term to identify a pattern or telescoping behavior.
2. Use partial fraction decomposition to rewrite the term. Set up the equation: \(\frac{2}{(2k - 1)(2k + 1)} = \frac{A}{2k - 1} + \frac{B}{2k + 1}\), and solve for constants \(A\) and \(B\).
3. After finding \(A\) and \(B\), express the general term as the difference of two simpler fractions, which will help in telescoping the series when summed from \(k=1\) to \(n\).
4. Write the partial sum \(S_n = \sum_{k=1}^n \frac{2}{(2k - 1)(2k + 1)}\) using the decomposed form, and observe how most terms cancel out when expanded.
5. Identify the remaining terms after cancellation to find a formula for \(S_n\), then take the limit as \(n \to \infty\) to conjecture the value of the infinite series.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. To find its value, we must determine if the series converges, meaning its partial sums approach a finite limit. Understanding convergence is essential before assigning a sum to an infinite series.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique helps simplify terms in the series, making it easier to identify patterns or telescoping behavior that can lead to a closed-form sum.

Telescoping Series

A telescoping series is one where many terms cancel out when partial sums are expanded. Recognizing telescoping allows us to simplify the sum of the series by focusing on the first and last terms of the partial sums, facilitating the evaluation of the infinite sum.