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


b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Verified step by step guidance

1. Start by writing the nth partial sum of the series as the sum of the first n terms: \(S_n = \sum_{k=1}^n \frac{2}{(2k - 1)(2k + 1)}\).
2. Use partial fraction decomposition to rewrite the general term \(\frac{2}{(2k - 1)(2k + 1)}\) in a form that will allow telescoping. Set \(\frac{2}{(2k - 1)(2k + 1)} = \frac{A}{2k - 1} + \frac{B}{2k + 1}\) and solve for A and B.
3. After finding A and B, rewrite each term of the sum using these partial fractions. This should create a telescoping series where most terms cancel out when summed.
4. Express \(S_n\) as a simplified expression after cancellation, which will be a formula involving only a few terms depending on n.
5. Use the formula for \(S_n\) to calculate the next four partial sums: \(S_5\), \(S_6\), \(S_7\), and \(S_8\) by substituting n = 5, 6, 7, and 8 respectively.

Key Concepts

Partial Sums of a Series

A partial sum Sₙ is the sum of the first n terms of a series. It helps approximate the value of an infinite series by adding a finite number of terms. Understanding partial sums is essential to analyze convergence and to find explicit formulas for sums up to a certain index.

Telescoping Series

A telescoping series is one where many terms cancel out when the partial sums are expanded. This simplification often allows finding a closed-form expression for Sₙ. Recognizing telescoping patterns is key to simplifying complex series like the given one.

Decomposition into Partial Fractions

Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique is useful to rewrite terms in a series to reveal telescoping behavior. Applying it to the given term helps in finding a formula for the nth partial sum.