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

Calculus: Early Transcendentals
8th Edition
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Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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EXAMPLE 9 Find the Maclaurin series for the function f(x)=√4=X
and its radius of convergence.
SOLUTION We write f(x) in a form where we can use the binomial
series.
1
4- X
11
II
=
Using the binomial series with k =
-x/4, we have
12/1₁
(
+
1
-1/2
00
√²+²=x = ²/(1-4) -¹/² =
1(4-1) **- 1 2 ( )
4 - X
n =
n
1
= — - [¹ + (− ² ) ( − 4 ) + ! ₂
2!
- (- -) ( - ²3 ) ( (
+
+
1
-1/2
and with x replaced by
3!
( - -³) (− ³ ) ( - ₹³) ·· · (-¯ ¼ ¯ ´ + ¹) (- ¼ )” + ···]
-)
-
n!
5x +
1.3.5...
n!8
-(-4) ²
D) (-4)³ -
1.3
218²
(2n-1)x+
+...
-x² +
1.3.5
318³
..].
+...
We know from The Binomial Series that this series converges when
|-x/4| < 1, that is |x| <
, so the radius of convergence is
R =
Transcribed Image Text:EXAMPLE 9 Find the Maclaurin series for the function f(x)=√4=X and its radius of convergence. SOLUTION We write f(x) in a form where we can use the binomial series. 1 4- X 11 II = Using the binomial series with k = -x/4, we have 12/1₁ ( + 1 -1/2 00 √²+²=x = ²/(1-4) -¹/² = 1(4-1) **- 1 2 ( ) 4 - X n = n 1 = — - [¹ + (− ² ) ( − 4 ) + ! ₂ 2! - (- -) ( - ²3 ) ( ( + + 1 -1/2 and with x replaced by 3! ( - -³) (− ³ ) ( - ₹³) ·· · (-¯ ¼ ¯ ´ + ¹) (- ¼ )” + ···] -) - n! 5x + 1.3.5... n!8 -(-4) ² D) (-4)³ - 1.3 218² (2n-1)x+ +... -x² + 1.3.5 318³ ..]. +... We know from The Binomial Series that this series converges when |-x/4| < 1, that is |x| < , so the radius of convergence is R =
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