71. Evaluating an infinite series two ways
Evaluate the series
∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.
a. Use a telescoping series argument.

Verified step by step guidance

1. Write out the general term of the series explicitly: \(a_k = \frac{4}{3^k} - \frac{4}{3^{k+1}}\).
2. Recognize that this is a telescoping series because each term can be rewritten to show cancellation between consecutive terms when summed.
3. Express the partial sum \(S_n = \sum_{k=1}^n \left( \frac{4}{3^k} - \frac{4}{3^{k+1}} \right)\) and write out the first few terms to observe the telescoping pattern.
4. Notice that most terms cancel out, leaving only the first term of the first fraction and the last term of the second fraction in the partial sum.
5. Write the simplified form of \(S_n\) after cancellation and then take the limit as \(n \to \infty\) to find the sum of the infinite series.

Key Concepts

Telescoping Series

A telescoping series is a series where many terms cancel out when the partial sums are expanded, leaving only a few terms to evaluate. This simplification makes it easier to find the sum of the infinite series by focusing on the first and last terms of the partial sums.

Infinite Geometric Series

An infinite geometric series has terms that multiply by a constant ratio each time. If the absolute value of the ratio is less than one, the series converges, and its sum can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio.

Partial Sums and Convergence

Partial sums are the sums of the first n terms of a series. Understanding how these sums behave as n approaches infinity helps determine if the series converges and what its sum is. For telescoping and geometric series, analyzing partial sums is key to evaluating the infinite sum.