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Find the radius of convergence and interval of convergence for ‡�(from n 1 to calculus write a power series for e to the 2x and find the interval of convergence 1 2. 1 n n n x. f x n. ∞ −. −. ∑ on its interval of convergence. (a) Find the interval Therefore the radius of convergence is. 1 . 2. When. 1. ,. 2 x − the series is is a geometric series that converges to. 1. 1 2x. for. 1 . 2 x Therefore. 2. 1. 1. Find the radius of convergence and interval of convergence of the following series. ∞. ∑ n 1. (−1)n (2x 2)n n2. 2. Find a power series representation for the  n 1. 1 n3/2 1. A) is a convergent p-series. B) is a divergent p series. C) converges by Problem 1 Find a power series for e−2x and detrmine its radius of convergence. Problem 2 1−3x2 then find the radius and interval of convergence. 2  March 20, 2015 Interval of Convergence for a Series Lets take a look at the series Find the partial sums for n 1 to n 7 when x 1.6 and when x 4 series are, and to compute the radius and the interval of convergence for a give cos(x) 1 . ) (2) ydy dx. 2x and y(1) 2. (Answer y2 2x2 2. ) (3) (3y2 2y) (19) an (1 . 3 n. )4n. (Ans convergent, use logarithm to find that limit is e12.). Find the radius of convergence and the interval of convergence of the series to multiply this out and look at the coefficients of the highest powers of n . In series convergance what matters is the sequence not the term. Then state and show which convergence/divergence test you used. (a) o n 1. 4n. 5n 1 Determine the radius and interval of convergence of the series given. (a) o n 1. (2x 一 5)n n lim n—o. (2x 一 5)n 1 n 1•n(2x 一 5)n t1 ¥ 一1t2x 一 5t1. 5. if∑Ÿn 0 cn(x - a)n 0 for all x in the interval of convergence, thencn 0 for all values of n. 6. n 0 ncnxn-Ù. (1) also determine the expression for 2xy, e.g.. 2xy Ÿ× n 0 have a power series within x - x0with a positive radius of convergence. 6 .5 .3 .2c0x. 6 . 2 2. 7 .6 .4 .3cÙx. 7 c0. . 1 . 2. 3 .2x s . 2 2. 6 .5 .3 .2x. Notice that, in writing out the term pertaining to n 0 in Equations 1 and 2, we have . Find the radius of convergence and interval of convergence of the series. A power series may represent a function f(x), in the sense that wherever the series converges, it converges to f(x). There are radius of convergence. Determine the interval of convergence for the series. ∞. ∑ n 0 xn. 3n. We also found that in some cases a power series represents a function Suppose f(x) an(x-c)n is a convergent power series with radius of Find the Taylor series centered at c 0 for f(x) x3 2x2 3x 4. If a function f has a Taylor series centered at c then the series converges in the largest interval  Find the radius of convergence and interval of convergence for each series. 1. ∞ n 1. (−2)nxn. 4. √ n. 5. ∞. ∑ n 1. (−1)n (x 2)n n2n. 6. ∞. ∑ n 1. (2x − 1)n. Radius and Interval of Convergence The convergence of the power series P ∞ n 0 c n(x−a)n is described by one of the three possibilities (a) There is a number R Exercise 18 Find the radius of convergence and interval of convergence of the series. ∞. ∑ n 0 n. 4n. (x 1)n. Solution We start by applying the Ratio Test. lim. For which values of x does the following series converge ∞. ∑ n 0. (2x)n ∞. ∑ n 0. 2nxn . interval of convergence as the series in the previous example. Also. 1. 1 x. 1 (−1)n 1nxn. (a) Find the radius and interval of convergence.

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