Combining power series Use the geometric series

f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,

to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.

f(x³) = 1/(1 − x³)

검증된 단계별 안내

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

To find the power series for \(f(x^{3}) = \frac{1}{1 - x^{3}}\), substitute \(x^{3}\) in place of \(x\) in the original series.

This gives \(f(x^{3}) = \sum_{k=0}^{\infty} (x^{3})^{k} = \sum_{k=0}^{\infty} x^{3k}\).

The power series representation is therefore \(\sum_{k=0}^{\infty} x^{3k}\), which is centered at 0.

Determine the interval of convergence by applying the original condition \(|x| < 1\) to the new variable: since the series is in terms of \(x^{3}\), the condition becomes \(|x^{3}| < 1\), which simplifies to \(|x| < 1\).

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주요 개념

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Geometric Series and Its Power Series Representation

A geometric series is a sum of the form ∑ x^k for k from 0 to infinity, which converges to 1/(1-x) when |x| < 1. This fundamental series allows us to express functions as infinite sums, facilitating manipulation and analysis of functions within their radius of convergence.

Substitution in Power Series

Substitution involves replacing the variable x in a known power series with another expression, such as x³. This transforms the original series into a new series representing a related function, while maintaining the structure of the series and adjusting the interval of convergence accordingly.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges. When substituting variables, the interval changes based on the new expression's magnitude. Determining this interval ensures the validity of the power series representation for the function.

Combining power series Use the geometric seriesf(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.f(x³) = 1/(1 − x³)