Textbook Question
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹
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15. Power Series
Introduction to Power Series
4:23 minutesProblem 11.2.60
Textbook Question
Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
f(x) = ln √(1 − x²)
Verified step by step guidance1
Rewrite the function to a simpler form using logarithm properties: \(f(x) = \ln \sqrt{1 - x^2} = \frac{1}{2} \ln(1 - x^2)\).
Recall the known power series expansion for \(\ln(1 - u)\) centered at 0, which is \(\ln(1 - u) = -\sum_{n=1}^{\infty} \frac{u^n}{n}\) for \(|u| < 1\).
Substitute \(u = x^2\) into the series to get \(\ln(1 - x^2) = -\sum_{n=1}^{\infty} \frac{x^{2n}}{n}\), valid for \(|x^2| < 1\) or \(|x| < 1\).
Multiply the entire series by \(\frac{1}{2}\) to match the original function: \(f(x) = \frac{1}{2} \ln(1 - x^2) = -\frac{1}{2} \sum_{n=1}^{\infty} \frac{x^{2n}}{n}\).
State the interval of convergence based on the substitution and original series: since \(|x| < 1\), the interval of convergence is \((-1, 1)\).
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Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series Representation
A power series is an infinite sum of terms in the form a_n(x - c)^n, where a_n are coefficients and c is the center. Representing functions as power series allows approximation and analysis using polynomials. Finding a power series for a function often involves manipulating known series or using derivatives and integrals.
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Intro to Power Series
Known Power Series and Manipulation
Common functions like ln(1+x), 1/(1-x), and sqrt(1-x) have established power series expansions. To find the series for a related function, we use substitution, algebraic manipulation, or differentiation/integration of these known series. For example, ln(√(1 - x²)) can be expressed using the series for ln(1 - x²).
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Interval of Convergence
The interval of convergence is the set of x-values for which the power series converges to the function. It depends on the radius of convergence and endpoint behavior. Determining this interval ensures the series accurately represents the function within that domain.
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