ISPA - Representation of Functions by Power Series

Representation of Functions by Power Series

image of Joseph Fourier Image Caption: Did You Know? The French mathematician Joseph Fourier (1768-1830) used power series to represent sine and cosine functions when trying to solve a problem related to heat conduction.

Functions Defined By Power Series

Recall from the previous lesson that the sum of a power series is a function f(x) = c₀ + c₁x + c₂x² + c₃x³ +... + c_nxⁿ +... whose domain is the series interval of convergence. Representing functions by power series is often very useful for integrating functions that do not have elementary antiderivatives, for solving differential equations, and for approximating functions by polynomials. Power series can be added and subtracted where their intervals of convergence intersect, and multiplied term by term, just as polynomials are added, subtracted, and multiplied. Convergent power series can be multiplied together, to form a new convergent series. When the factors both converge, the product also converges. Operations with power series conform to the rules outlined below

Geometric Power Series

One technique for finding a power series that represents a given function relies on the relationship between a geometric series sum and its related function. It is known that the geometric series equation image indicator $LaTeX: \sum_{n=0}^\infty x^n=1+x+x^2+... = \frac{1}{1-x}$ $\sum_{n=0}^\infty x^n=1+x+x^2+... = \frac{1}{1-x}$ converges for $LaTeX: \left|x\right|<1$ $\left|x\right|<1$ . By replacing x with -x, the series becomes $LaTeX: \sum_{n=0}^\infty (-1)^nx^n=1+x+x^2-x^3... = \frac{1}{1+x}$ $\sum_{n=0}^\infty (-1)^nx^n=1+x+x^2-x^3... = \frac{1}{1+x}$ , which is a convergent alternating series on the interval (-1,1). Replacing x with -x² produces equation image indicator $LaTeX: \sum_{n=0}^\infty (-1)^nx^{2n}=1+x-x^2+x^4-x^6+... = \frac{1}{1+x^2}$ $\sum_{n=0}^\infty (-1)^nx^{2n}=1+x-x^2+x^4-x^6+... = \frac{1}{1+x^2}$ , which also converges on the interval (-1, 1). The next two presentations provide specific criteria and guidance for representing a function as a geometric power series.

Differentiation and Integration of Power Series

Other techniques for finding a power series to represent a known function include differentiation and anti-differentiation. A power series may be differentiated term by term at each interior point of its interval of convergence and integrated term by term throughout its interval of convergence. The Term-by-Term Differentiation Theorem and the Term-byTerm Integration Theorem guarantee this is true.

description of term-by-term differentiation theorem description of term-by-term integration theorem

The examples below illustrate how differentiation may be used to determine a function from a power series and how integration may used to determine a power series from a function.

Representation of Functions by Power Series Practice

Representation of Functions by Power Series: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of representation of functions as power series.

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