Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [

illusiia

illusiia

Answered question

2021-10-19

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 Rn(x)0.] Also find the associated radius of convergence. f(x)=2x

Answer & Explanation

Alix Ortiz

Alix Ortiz

Skilled2021-10-20Added 109 answers

Find a few derivatives, and calculate their values at a=0.
f(x)=2x f(0)=1
f(x)=(ln2)2x f(0)=ln2
f(x)=(ln2)22x f(0)=(ln2)2
f(x)=(ln2)32x f(0)=(ln2)3
f(4)(x)=(ln2)42x f(4)(0)=(ln2)4
Plug everything into the Maclaurin general form
f(x)=f(0)+f(0)1!x+f(0)2!x2+f(0)3!x3+
f(x)=1+ln21!x+(ln2)22!x2+(ln2)33!x3+(ln2)44!x4+
Find the pattern of the numbers to erite in summation form
f(x)=n=0(ln2)nxnn!
Use the Ratio Test
an=(ln2)nxnn!
|an+1an|=|(ln2)n+1xn+1(n+1)!n!(ln2)nxn|=|(ln2)xn+1|

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