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.

g(x) = x³/(1 − x)

검증된 단계별 안내

Recall the given geometric series formula: \[f(x) = \frac{1}{1 - x} = \sum_{k=0}^{\infty} x^{k}, \quad \text{for } |x| < 1.\] This series converges when the absolute value of \(x\) is less than 1.

To find the power series representation for \[g(x) = \frac{x^{3}}{1 - x},\] notice that it can be written as \[g(x) = x^{3} \cdot \frac{1}{1 - x}.\]

Substitute the power series for \[\frac{1}{1 - x}\] into the expression for \[g(x)\]: \[g(x) = x^{3} \sum_{k=0}^{\infty} x^{k}.\]

Use properties of exponents to combine the powers of \[x\] inside the summation: \[g(x) = \sum_{k=0}^{\infty} x^{k + 3} = \sum_{k=0}^{\infty} x^{k + 3}.\]

Rewrite the series with a new index if desired (for example, let \[n = k + 3\]) to express the series in standard form. The interval of convergence remains \[|x| < 1\] because multiplying by \[x^{3}\] does not affect convergence.

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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 many functions as power series by manipulating the variable or coefficients. Understanding this series is key to rewriting functions like g(x) in terms of power series.

Manipulating Power Series to Represent New Functions

To find the power series of a function like g(x) = x³/(1 - x), we use substitution and multiplication on the known geometric series. Multiplying the entire series by x³ shifts the powers, resulting in a new series ∑ x^(k+3). This technique helps generate power series for more complex functions based on simpler known series.

Interval of Convergence for Power Series

The interval of convergence is the set of x-values for which a power series converges. For the geometric series 1/(1 - x), the interval is |x| < 1. When manipulating the series (e.g., multiplying by x³), the interval of convergence remains the same unless the function introduces new singularities. Identifying this interval ensures the series representation is valid.

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.g(x) = x³/(1 − x)