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

b. Use a geometric series argument with Theorem 10.8.

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First, rewrite the given series to clearly see its terms: \( \sum_{k=1}^{\infty} \left( \frac{4}{3^k} - \frac{4}{3^{k+1}} \right) \). Notice that each term is a difference of two fractions involving powers of 3.

Next, separate the series into two separate sums: \( \sum_{k=1}^{\infty} \frac{4}{3^k} - \sum_{k=1}^{\infty} \frac{4}{3^{k+1}} \). This will help us analyze each sum individually.

Recognize that both sums are geometric series. The first sum has the first term \( a_1 = \frac{4}{3} \) and common ratio \( r = \frac{1}{3} \). The second sum starts from \( k=1 \) but with terms \( \frac{4}{3^{k+1}} \), which can be rewritten as \( \frac{4}{3} \cdot \left( \frac{1}{3} \right)^k \).

Apply the formula for the sum of an infinite geometric series, which is \( S = \frac{a_1}{1 - r} \), to both sums separately. For the first sum, use \( a_1 = \frac{4}{3} \) and \( r = \frac{1}{3} \). For the second sum, identify the first term and ratio accordingly and apply the formula.

Finally, subtract the sum of the second series from the sum of the first series to find the value of the original series. This step uses the linearity of summation and the geometric series sums derived.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. To evaluate such a series, it is crucial to determine whether it converges to a finite value. Convergence depends on the behavior of the terms as the index approaches infinity, and only convergent series have meaningful sums.

Geometric Series and Its Sum Formula

A geometric series is a series where each term is a constant multiple (common ratio) of the previous term. The sum of an infinite geometric series with |r| < 1 is given by S = a / (1 - r), where a is the first term. This formula allows quick evaluation of many infinite series.

Telescoping Series Technique

A telescoping series is one where many terms cancel out when the series is expanded, simplifying the sum. Recognizing the series as telescoping helps evaluate it by focusing on the first few and last few terms. This technique often complements geometric series arguments.

71. Evaluating an infinite series two waysEvaluate the series∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.b. Use a geometric series argument with Theorem 10.8.