Math Problem Statement

Find the power series representation for g centered at 0 by differentiating or integrating the power series for f​ (perhaps more than​ once). Give the interval of convergence for the resulting series. ​g(x)equalsStartFraction x Over left parenthesis 1 plus 11 x squared right parenthesis squared EndFraction using ​f(x)equalsStartFraction 1 Over 1 plus 11 x squared EndFraction Question content area bottom Part 1 ​g(x)equalsSummation from k equals 0 to infinity    enter your response here

Solution

To find the power series representation of g(x)=x(1+11x2)2g(x) = \frac{x}{(1 + 11x^2)^2} by using f(x)=11+11x2f(x) = \frac{1}{1 + 11x^2}, we start by expressing f(x)f(x) as a power series and then differentiate it as needed to obtain g(x)g(x).

Step 1: Express f(x)=11+11x2f(x) = \frac{1}{1 + 11x^2} as a Power Series

We know that: 11u=n=0un\frac{1}{1 - u} = \sum_{n=0}^{\infty} u^n for u<1|u| < 1.

Here, we can set u=11x2u = -11x^2, so: f(x)=11+11x2=n=0(11x2)nf(x) = \frac{1}{1 + 11x^2} = \sum_{n=0}^{\infty} (-11x^2)^n Expanding this series: f(x)=n=0(11)nx2nf(x) = \sum_{n=0}^{\infty} (-11)^n x^{2n}

Step 2: Differentiate f(x)f(x) to Find g(x)g(x)

Since g(x)=x(1+11x2)2=xf(x)g(x) = \frac{x}{(1 + 11x^2)^2} = x \cdot f'(x), we need to differentiate f(x)f(x) with respect to xx.

f(x)=n=0(11)nx2nf(x) = \sum_{n=0}^{\infty} (-11)^n x^{2n} Differentiating term by term: f(x)=n=1(11)n2nx2n1f'(x) = \sum_{n=1}^{\infty} (-11)^n \cdot 2n x^{2n-1} Now, multiply by xx to find g(x)g(x): g(x)=xf(x)=n=1(11)n2nx2ng(x) = x \cdot f'(x) = \sum_{n=1}^{\infty} (-11)^n \cdot 2n x^{2n}

Final Answer and Interval of Convergence

Thus, the power series representation for g(x)g(x) is: g(x)=n=1(11)n2nx2ng(x) = \sum_{n=1}^{\infty} (-11)^n \cdot 2n x^{2n}

The interval of convergence is determined by the original series for f(x)f(x), which converges for 11x2<1|11x^2| < 1, or x<111|x| < \frac{1}{\sqrt{11}}.

Summary

  1. Power series for g(x)g(x): g(x)=n=1(11)n2nx2ng(x) = \sum_{n=1}^{\infty} (-11)^n \cdot 2n x^{2n}
  2. Interval of Convergence: x<111|x| < \frac{1}{\sqrt{11}}

Would you like additional details on the steps, or have any questions?

Related Questions

  1. How would you find the power series representation for a different function related to f(x)f(x)?
  2. What if the center of convergence was at a different point? How would the series change?
  3. Could this method be applied to similar rational functions, and what might differ?
  4. How would the radius of convergence change if the function had a different constant instead of 11?
  5. What are alternative methods for finding the power series of g(x)g(x) besides differentiation?

Tip: For power series, identifying the core structure, like 11u\frac{1}{1 - u}, is crucial for expanding complex functions into a series form efficiently.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Power Series
Differentiation of Series
Interval of Convergence

Formulas

Geometric series: 1 / (1 - u) = sum(u^n) for |u| < 1
Differentiation of power series term-by-term

Theorems

Power Series Convergence Theorem
Term-by-Term Differentiation Theorem

Suitable Grade Level

Undergraduate Calculus