Represent the function 3ln(8-x) as a power series (Maclaurin series) What is the radius of convergence?

Khadija Wells 2021-02-01 Answered
Represent the function 3ln(8x) as a power series (Maclaurin series)
What is the radius of convergence?
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Expert Answer

yunitsiL
Answered 2021-02-02 Author has 108 answers

To write the function f(x)=3ln(8x) as a power series (Maclaurin series). Also to calculate the radius of convergence.
From the standard power series, the Maclaurin series of ln(1+x) is given by,
ln(1+x)=n=1(1)n1xnn, |x|<1
Here,
|x|<1 is its radius of converhence
Rewrite the function as shown below
f(x)=3ln(8x)
f(x)=3ln(8(1x8))
f(x)=3ln(8)+3ln(1x8)
Now, ln(1x8)=n=1(1)n1(x8)nn,|x8|<1
ln(1x8)=n=1(1)n1(1)nxnn8n|x|8<1
ln(1x8)=n=1(1)2nxnn8n|x|<8
ln(1x8)=n=1xnn8n|x|<8
Use it above
f(x)=3ln(8)3n=1xnn8n
This is the required power series and the radius of convergence is 8 (from |x|<8)
Answer:
Power series: f(x)=3ln(8)3n=1xnn8n
The radius of convergence is 8.

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Jeffrey Jordon
Answered 2021-12-16 Author has 2047 answers

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