Question
Asked Dec 3, 2019
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Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R(x) 0.]
f(x) sin(x)
C0
Σ
f(x)=
n 0
Find the associated radius of convergence R.
R =
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Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R(x) 0.] f(x) sin(x) C0 Σ f(x)= n 0 Find the associated radius of convergence R. R =

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Expert Answer

Step 1

Recall the definition of a Maclaurin series as follows.

*...о
f(0)x f"(0)
(0)
х" +--
ГО)x ,0) (0),
f(x)f(0)+
1!
+
2!
3!
п!
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*...о f(0)x f"(0) (0) х" +-- ГО)x ,0) (0), f(x)f(0)+ 1! + 2! 3! п!

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Step 2

Evaluate the values of derivatives of sin(x) at x = 0 as follows.

f(x)sinx
f(0)=sin (0)0
f(0)-[cos.xcos(0)-1
f'(0)[-sinr =-sin(0)= 0
(0)cos.x-cos(0) =-
f(0)-sinrsin (0) 0
f(0)=[cosx]=cos (0) = 1
x-0
The coefficients alternate between 0,1,-1
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f(x)sinx f(0)=sin (0)0 f(0)-[cos.xcos(0)-1 f'(0)[-sinr =-sin(0)= 0 (0)cos.x-cos(0) =- f(0)-sinrsin (0) 0 f(0)=[cosx]=cos (0) = 1 x-0 The coefficients alternate between 0,1,-1

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Step 3

Obtain the Maclaurin se...

(0)+
f(x) f(0)+()xf"(0)f0)
(0)
1!
2!
3!
п!
(-1)
x0
1
5
=0+ +0+
1!
3!
5!
= x
3!
5!
(-1)"x2
O(2n 1)!
2n+1
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(0)+ f(x) f(0)+()xf"(0)f0) (0) 1! 2! 3! п! (-1) x0 1 5 =0+ +0+ 1! 3! 5! = x 3! 5! (-1)"x2 O(2n 1)! 2n+1

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Advanced Math