Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| \u003c 1, to determine the Maclaurin series and the interval of convergence for the following functions.ƒ(x) = 1/(1 - x²)

Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 11 - Power Series

문제 11.R.29

11장, 문제 11.R.29

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.

ƒ(x) = 1/(1 - x²)

검증된 단계별 안내

Recall the geometric series formula: \(\sum_{k=0}^{\infty} x^k = \frac{1}{1 - x}\) for \(|x| < 1\).

Notice that the given function is \(f(x) = \frac{1}{1 - x^2}\), which can be seen as \(\frac{1}{1 - (x^2)}\).

Apply the geometric series formula by substituting \(x\) with \(x^2\), so the series becomes \(\sum_{k=0}^{\infty} (x^2)^k = \sum_{k=0}^{\infty} x^{2k}\).

Write the Maclaurin series for \(f(x)\) as \(\sum_{k=0}^{\infty} x^{2k}\), which expands to \(1 + x^2 + x^4 + x^6 + \cdots\).

Determine the interval of convergence by applying the condition \(|x^2| < 1\), which simplifies to \(|x| < 1\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Geometric Series

A geometric series is an infinite sum of the form Σ x^k from k=0 to ∞, which converges to 1/(1 - x) when |x| < 1. This formula provides a foundation for expressing functions as power series by recognizing patterns similar to geometric series.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series centered at zero, representing a function as an infinite sum of its derivatives at zero multiplied by powers of x. It allows us to express functions as power series expansions around x = 0.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to the function it represents. For geometric series, this interval is determined by the condition |x| < 1, and it must be adjusted when the series involves expressions like x².

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.ƒ(x) = 1/(1 - x²)