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 by using , we start by expressing as a power series and then differentiate it as needed to obtain .
Step 1: Express as a Power Series
We know that: for .
Here, we can set , so: Expanding this series:
Step 2: Differentiate to Find
Since , we need to differentiate with respect to .
Differentiating term by term: Now, multiply by to find :
Final Answer and Interval of Convergence
Thus, the power series representation for is:
The interval of convergence is determined by the original series for , which converges for , or .
Summary
- Power series for :
- Interval of Convergence:
Would you like additional details on the steps, or have any questions?
Related Questions
- How would you find the power series representation for a different function related to ?
- What if the center of convergence was at a different point? How would the series change?
- Could this method be applied to similar rational functions, and what might differ?
- How would the radius of convergence change if the function had a different constant instead of 11?
- What are alternative methods for finding the power series of besides differentiation?
Tip: For power series, identifying the core structure, like , is crucial for expanding complex functions into a series form efficiently.
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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
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