Textbook Question
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]
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15. Power Series
Introduction to Power Series
5:38 minutesProblem 11.2.61
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 √(4 − x²)
Verified step by step guidance1
Rewrite the function to a more convenient form: \( f(x) = \ln \sqrt{4 - x^2} = \frac{1}{2} \ln(4 - x^2) \). This simplifies the problem to finding the power series for \( \ln(4 - x^2) \).
Express \( \ln(4 - x^2) \) in terms of \( \ln 4 \) and a logarithm of a form suitable for a known power series: \( \ln(4 - x^2) = \ln 4 + \ln\left(1 - \frac{x^2}{4}\right) \).
Recall the known power series for \( \ln(1 - u) = -\sum_{n=1}^\infty \frac{u^n}{n} \) valid for \( |u| < 1 \). Here, set \( u = \frac{x^2}{4} \).
Substitute \( u = \frac{x^2}{4} \) into the series to get \( \ln\left(1 - \frac{x^2}{4}\right) = -\sum_{n=1}^\infty \frac{1}{n} \left(\frac{x^2}{4}\right)^n = -\sum_{n=1}^\infty \frac{x^{2n}}{n 4^n} \).
Combine all parts to write the power series for \( f(x) \): \( f(x) = \frac{1}{2} \ln 4 - \frac{1}{2} \sum_{n=1}^\infty \frac{x^{2n}}{n 4^n} \). The interval of convergence comes from \( |u| = \left|\frac{x^2}{4}\right| < 1 \), which simplifies to \( |x| < 2 \).
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Video duration:
5mKey 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 c is the center of the series. 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, express it in terms of these known forms and apply algebraic operations, substitutions, or differentiation/integration to derive the new series.
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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, often found using the ratio or root test. Determining this interval is crucial to ensure the series accurately represents the function within that domain.
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