EXAMPLE 9 Find the Maclaurin series for the function f(x)=√4=Xand its radius of convergence.SOLUTION We write f(x) in a form where we can use the binomialseries.14- X11II=Using the binomial series with k =-x/4, we have12/1₁(+1-1/200√²+²=x = ²/(1-4) -¹/² =1(4-1) **- 1 2 ( )4 - Xn =n1= — - [¹ + (− ² ) ( − 4 ) + ! ₂2!- (- -) ( - ²3 ) ( (++1-1/2and with x replaced by3!( - -³) (− ³ ) ( - ₹³) ·· · (-¯ ¼ ¯ ´ + ¹) (- ¼ )” + ···]-)-n!5x +1.3.5...n!8-(-4) ²D) (-4)³ -1.3218²(2n-1)x++...-x² +1.3.5318³..].+...We know from The Binomial Series that this series converges when|-x/4| < 1, that is |x| <, so the radius of convergence isR =

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EXAMPLE 9 Find the Maclaurin series for the function f(x) and its radius of convergence. 1 4- X SOLUTION We write f(x) in a form where we can use the binomial series. || 1 √ 4²-x = ²/(1-4)-¹/² X || 1 ( Using the binomial series with k = -x/4, we have 1 1 ¹-√ ₁ + (-²¹) (-) · 1 + -1/2 00 = + 2 (-³) ( Σ 2 n = 0 n 1 4 and with x replaced by 2 (-x)²

EXAMPLE 9 Find the Maclaurin series for the function f(x) and its radius of convergence. 1 4- X SOLUTION We write f(x) in a form where we can use the binomial series. || 1 √ 4²-x = ²/(1-4)-¹/² X || 1 ( Using the binomial series with k = -x/4, we have 1 1 ¹-√ ₁ + (-²¹) (-) · 1 + -1/2 00 = + 2 (-³) ( Σ 2 n = 0 n 1 4 and with x replaced by 2 (-x)²

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 67E

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