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


a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Verified step by step guidance

1. Identify the general term of the series: \( a_k = \frac{2}{(2k - 1)(2k + 1)} \).
2. Use partial fraction decomposition to rewrite \( a_k \) in a form that allows telescoping. Set \( \frac{2}{(2k - 1)(2k + 1)} = \frac{A}{2k - 1} + \frac{B}{2k + 1} \) and solve for constants \( A \) and \( B \).
3. Express each partial sum \( S_n = \sum_{k=1}^n a_k \) by substituting the decomposed form of \( a_k \) and write out the sum explicitly to observe cancellation of terms.
4. Calculate the first four partial sums \( S_1, S_2, S_3, S_4 \) by summing the first 1, 2, 3, and 4 terms respectively, using the telescoping form to simplify the sums.
5. Write each partial sum \( S_n \) in its simplified form after cancellation, which will help in understanding the behavior of the series as \( n \) increases.

Key Concepts

Partial Sums of a Series

Partial sums are the sums of the first n terms of a series, denoted as Sₙ = a₁ + a₂ + ... + aₙ. They help analyze the behavior of infinite series by approximating the total sum and are essential for understanding convergence and series evaluation.

Telescoping Series

A telescoping series is one where many terms cancel out when partial sums are expanded, simplifying the sum significantly. Recognizing telescoping patterns allows easier computation of partial sums and limits, often by expressing terms as differences of fractions.

Decomposition into Partial Fractions

Partial fraction decomposition breaks a complex rational expression into simpler fractions that are easier to sum or integrate. In series, this technique often reveals telescoping behavior by rewriting terms to highlight cancellations.