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

검증된 단계별 안내

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 \( r = -\frac{2}{7} \).

Write the formula for the nth partial sum \( S_n \) of a geometric series: \[ S_n = a \frac{1 - r^{n+1}}{1 - r} \]

Substitute the values of \( a \) and \( r \) into the formula to express \( S_n \) explicitly: \[ S_n = 1 \times \frac{1 - \left(-\frac{2}{7}\right)^{n+1}}{1 - \left(-\frac{2}{7}\right)} \]

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: \[ \lim_{n \to \infty} S_n = \frac{1}{1 - \left(-\frac{2}{7}\right)} \]

Simplify the expression for the limit to find the sum of the infinite geometric series.

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주요 개념

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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.

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