Find the Taylor series centered at zero for the function f(x)=ln(2 + x^2). Determine the radius of convergence of this series.

BenoguigoliB 2020-12-15 Answered
Find the Taylor series centered at zero for the function f(x)=ln(2+x2). Determine the radius of convergence of this series.
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Expert Answer

SkladanH
Answered 2020-12-16 Author has 80 answers

Find the Taylor series centered at zero for the function f(x)=ln(2+x2). Determine the radius of convergence of this series.
Consider a given function is
f(x)=ln(2+x2).
Now,
f(x)=ln[2(1+x22)]
=ln(2)+ln(1+x22)
=ln(2)+n=1(1)n+1(x22)nn...[ln(1+x)=n=1(1)n+1xnn]
=ln(2)+n=1(1)n+1x2nn2n
Hence, the required Taylor's series of f(x)=ln(2+x2) is
ln(2)+n=1(1)n+1x2nn2n
Consider a Taylor's series is ln(2)+n=1(1)n+1x2nn2n
To find a Taylor's series, consider a sequence <(1)n+1x2nn2n>
Now, by ratio test,
L=limn|an+1an|
=limn|(1)n+1x2n+2(n+1)2n+1(1)n+1x2nn2n|
=limn|n2nx2n+2(n+1)2n+1x2n|
=limn|nx2(n+1)2|
=|x22|

The series is convergent if L<1. Hence, |x22|<1

It implies that 0

Check at the end point x=2. Then,

n=1(1)n+1x2nn2n=n=1(1)n+1(2)2nn2n

=n=1(1)n+1n

It is convergent series.

Hence, the radius of convergence is 0

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Jeffrey Jordon
Answered 2021-12-17 Author has 2070 answers

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