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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.4.55
Chapter 11, Problem 11.4.55

Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ (xᵏ)/(2ᵏ)

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1
Recognize that the given series is a geometric series of the form \(\sum_{k=0}^\infty ar^k\), where \(a\) is the first term and \(r\) is the common ratio.
Identify the first term \(a\) by substituting \(k=0\) into the series: \(a = \frac{x^0}{2^0} = 1\).
Identify the common ratio \(r\) by looking at the general term: \(r = \frac{x}{2}\).
Recall the formula for the sum of an infinite geometric series when \(|r| < 1\): \(S = \frac{a}{1 - r}\).
Write the function represented by the power series as \(f(x) = \frac{1}{1 - \frac{x}{2}}\) and simplify if desired.

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Key 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_k(x - c)^k, where a_k are coefficients and c is the center. Understanding how functions can be expressed as power series allows us to analyze and approximate functions within their radius of convergence.
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Intro to Power Series

Geometric Series and Its Sum

A geometric series is a power series where each term is a constant ratio times the previous term, ∑ r^k. Its sum converges to 1/(1 - r) when |r| < 1. Recognizing a power series as geometric helps identify the function it represents.
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Radius and Interval of Convergence

The radius of convergence defines the interval around the center where the power series converges to a finite value. For the series ∑ (x^k)/(2^k), convergence depends on |x/2| < 1, which is essential to determine the domain where the function representation is valid.
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Radius of Convergence
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