Power series play a pivotal role in calculus, enabling the differentiation and integration of functions with ease. They are infinite sums centered at a point, used to express complex functions as simpler ones. This text delves into the geometric series as a power series example, convergence determination using the radius and interval, and the representation of functions like exponential and trigonometric functions through power series. Understanding these series is key to solving a wide range of calculus problems.
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Power series are infinite sums of the form \( \sum _{n=0} ^{\infty} c_{n} (x-a) ^{n} \) used in calculus for differentiation and integration
Convergence of Geometric Series
The geometric series \( \sum _{n=0} ^{\infty} ar ^{n} \) converges when \( |r|<1 \) and sums to \( \frac{1}{1-x} \) for \( |x|<1 \)
Importance of Geometric Series
The geometric series is essential for understanding the convergence behavior of power series and computing their sums
The radius of convergence, \( R \), is the distance from the center \( a \) within which the power series converges, and the interval of convergence is the set of all \( x \) values for which the series converges
The Ratio Test is used to determine the convergence of a power series by comparing the ratio of consecutive terms to a limit
The Root Test is used to determine the convergence of a power series by comparing the nth root of the absolute value of the terms to a limit
The series \( \sum _{n=0} ^{\infty} \frac{x ^{n}}{n!} \) has an infinite radius of convergence and an interval of convergence of \( (-\infty, +\infty) \), while \( \sum _{n=0} ^{\infty} \frac{(x-1)^{n}}{n+1} \) has a radius of convergence \( R=1 \) and an interval of convergence of \( [0, 2) \)
Functions can be expressed as power series within a certain radius of convergence, making them useful for representing complex functions
The derivative of a power series is obtained by applying the power rule to each term, allowing for the differentiation of functions such as \( e^x \)
Power series for \( \sin(x) \), \( \cos(x) \), and \( e^x \) are commonly used in calculus and have infinite radii of convergence
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Ratio Test for Series Convergence
Compares the limit of absolute value of consecutive terms. If limit < 1, series converges; if > 1, diverges; if = 1, test is inconclusive.
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Radius of Convergence Concept
Distance from the center of a power series where the series converges. Found using Ratio Test or other convergence tests.
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Interval of Convergence Evaluation
Includes all x-values for which a power series converges. Found by testing endpoints after determining radius with Ratio Test.
