Lecture 8-7 - | x-a | < R (i.e. take everything that’s...

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8.7 - Power Series Defn. Power Series A power series about x = a is a series of the form where the c i ’s are constants (but can be functions of n ). Examples of Power Series 1. X n =0 ( - 1) n +1 ( x + 2) n n ! 2. X n =0 x n 2 n Question: For what values of x do power series converge? Answer: To answer this fully, we’ll need to first introduce some terminology. Note: 1
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Radius and Interval of Convergence of a Power Series Defn. Radius of Convergence and Interval of Convergence Suppose X n =0 c n ( x - a ) n converges for | x - a | < R . R is called the radius of convergence . The interval I on which the series converges is called the interval of convergence . How to find the radius of convergence R Consider the power series X n =0 c n ( x - a ) n . To find R : 1. Calculate the limit of the Ratio Test on the absolute value of the terms of the series. 2. For convergence, the Ratio Test requires L < 1. This gives an inequality involving | x - a | . Solve the inequality so that it is in the form
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Unformatted text preview: | x-a | < R (i.e. take everything that’s not | x-a | to the other side, this includes constants). You’ve found R ! 2 How to find the interval of convergence I 1. Find R first, i.e. the equation | x-a | < R . 2. Expand the inequality. 3. Test the series at the endpoints to see if the series converges there. Examples Find the radius and interval of convergence of the following series. 1. ∞ X n =0 1 n + 1 ( x-4) n 3 2. ∞ X n =0 ( x-2) n 10 n 3. ∞ X n =0 3 n x n n ! 4 4. ∞ X n =0 n !( x + 2) n In summary, there are 3 possibilities when finding R : 1. ∑ converges only when x = a . 2. ∑ converges for all x . 3. There is an R > 0 such that ∑ converges if | x-a | < R and diverges if | x-a | > R . 5...
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This note was uploaded on 03/03/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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Lecture 8-7 - | x-a | < R (i.e. take everything that’s...

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