# 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)C0Σf(x)=n 0Find the associated radius of convergence R.R =

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1 views help_outlineImage TranscriptioncloseFind 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 = fullscreen
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Step 1

Recall the definition of a Maclaurin series as follows. help_outlineImage Transcriptionclose*...о f(0)x f"(0) (0) х" +-- ГО)x ,0) (0), f(x)f(0)+ 1! + 2! 3! п! fullscreen
Step 2

Evaluate the values of derivatives of sin(x) at x = 0 as follows. help_outlineImage Transcriptionclosef(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 fullscreen
Step 3

Obtain the Maclaurin se... help_outlineImage Transcriptionclose(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 fullscreen

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