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


a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.


∑ (k = 0 to ∞) (–2/7)ᵏ

Verified step by step guidance

1. Identify the first term \( a \) and the common ratio \( r \) of the geometric series. Here, \( a = 1 \) (since when \( k=0 \), \( (-2/7)^0 = 1 \)) and \( r = -\frac{2}{7} \).
2. Write the formula for the nth partial sum \( S_n \) of a geometric series: \[ S_n = a \frac{1 - r^{n+1}}{1 - r} \]
3. Substitute the values of \( a \) and \( r \) into the formula to express \( S_n \) explicitly: \[ S_n = 1 \times \frac{1 - \left(-\frac{2}{7}\right)^{n+1}}{1 - \left(-\frac{2}{7}\right)} \]
4. Evaluate the limit of \( S_n \) as \( n \to \infty \). Since \( |r| = \frac{2}{7} < 1 \), the term \( r^{n+1} \) approaches zero, so the limit is: \[ \lim_{n \to \infty} S_n = \frac{1}{1 - \left(-\frac{2}{7}\right)} \]
5. Simplify the expression for the limit to find the sum of the infinite geometric series.

Key Concepts

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding this structure is essential for finding partial sums and limits.

Partial Sum of a Geometric Series

The nth partial sum Sₙ of a geometric series is the sum of the first n+1 terms. It can be calculated using the formula Sₙ = a(1 - r^(n+1)) / (1 - r) when r ≠ 1. This formula helps in evaluating the series up to a finite number of terms.

Limit of an Infinite Geometric Series

If the absolute value of the common ratio |r| < 1, the infinite geometric series converges, and its sum is the limit of the partial sums as n approaches infinity. This limit is given by S = a / (1 - r), providing the sum of infinitely many terms.