18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.


b. Evaluate the series using Theorem 10.7.


∑ (k = 0 to ∞) (–2/7)ᵏ

Verified step by step guidance

1. Identify the geometric series given: \(\sum_{k=0}^{\infty} \left(-\frac{2}{7}\right)^k\). Here, the first term \(a\) is when \(k=0\), which is \(\left(-\frac{2}{7}\right)^0 = 1\).
2. Recognize the common ratio \(r\) of the geometric series, which is the base of the exponent: \(r = -\frac{2}{7}\).
3. Check the convergence condition for an infinite geometric series: the series converges if and only if \(|r| < 1\). Since \(| -\frac{2}{7} | = \frac{2}{7} < 1\), the series converges.
4. Apply Theorem 10.7 for the sum of an infinite geometric series, which states that if \(|r| < 1\), then the sum \(S\) is given by \(S = \frac{a}{1 - r}\).
5. Substitute the values of \(a = 1\) and \(r = -\frac{2}{7}\) into the formula to express the sum as \(S = \frac{1}{1 - \left(-\frac{2}{7}\right)}\). Simplify the denominator to get the final expression for the sum.

Key Concepts

Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It can be finite or infinite, and its general form is ∑ ar^k, where a is the first term and r is the common ratio.

Convergence of Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r), providing a finite value for the infinite sum.

Theorem 10.7 (Sum of an Infinite Geometric Series)

Theorem 10.7 states that for an infinite geometric series with first term a and common ratio r, if |r| < 1, then the sum is S = a / (1 - r). This theorem provides a direct method to evaluate infinite geometric series without summing individual terms.