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Notes on the Interval of Convergence for a Power Series

# Notes on the Interval of Convergence for a Power Series -...

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Unformatted text preview: Notes on the Interval of Convergence for a Power Series Definition : Let ( 29 n n n c x a ∞ =- ∑ denote a power series about x a = . The Interval of Convergence (sometimes denoted by IOC ) for ( 29 n n n c x a ∞ =- ∑ is the set of all values of x for which the power series converges. It is related to the ROC by the following where R is the ROC for ( 29 n n n c x a ∞ =- ∑ : o If R = ∞ , then the interval of convergence is ( 29 ,-∞ ∞ or, equivalently, x-∞ < < ∞ . o If R = , then the interval of convergence is the singleton set { } a . o If0 R < < ∞ , then the series converges for x a R- < , or R x a R- <- < ⇔ a R x a R- < < + . So the interval of convergence contains the interval ( 29 , a R a R- + together with any endpoints for which the power series converges. Examples: Below the examples form the previous class are examined to determine their respective intervals of convergence. The answers are based on the previous work in determining the radius of convergence. In order to find the interval of convergence for a power series, one first finds the radius of convergence. 1) Consider the power series n n x ∞ = ∑ : Since this power series is the Geometric Series, we know that the series converges if and only if 1 x < . So the interval of convergence is 1 x < , or equivalently, 1 1 x- < < . 2) Consider the power series ( 29 1 ! n n x n ∞ =- ∑ : The radius of convergence for the power series is R = ∞ . So the interval of convergence is ( 29 ,-∞ ∞ , or equivalently, x-∞ < < ∞ . 3) Consider the power series ( 29 ! 3 n n n x ∞ = + ∑ : The radius of convergence for the power series is zero. So the interval of convergence is the singleton set { } 3- ....
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Notes on the Interval of Convergence for a Power Series -...

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