Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x)=lnsqrt{4-x}

CoormaBak9 2020-10-28 Answered
Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
f(x)=ln4x
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Expert Answer

l1koV
Answered 2020-10-29 Author has 100 answers

Given, f(x)=ln4x
Find power series representation, the radius of converges, the interval of converges.
Use the power series formula,
 f(x)=ln(1xn)=k=1xkk,1x<1

Consider
f(x)=ln4x2
f(x)=ln(4x2)12
Use the power rule of logarithmic,
lnm2=nlnm
=12ln(4x2)
Now multiple and divide "4" on the logarithmic function we get,
=12ln4(4x24)
=12ln4(44x24)
=12ln4(1x24)
Now use the product rule of logarithmic we get,
ln(mn)=ln(m)+ln(n)
=12ln4+ln(1x24)
Now again use the power rule of logarithmic we get,
=ln412+ln(1x24)
=ln222+ln(1x24)
=ln2+ln(1x24)
Now use the power series formula we get,
=ln2k=1(x24)kk
=ln2k=1(x2k4k)k
For this we get interval, 2x2
Use Ratio test to check the series converges or diverges:
Ratio test definition:
Suppose we have series an defined,
L=limn|an+1an|
If L>1 then the series diverges,
If L<1 then the series converges,
If L=1 then the test fails.
Consider,
ak=x2k4kk
ak+1=x2(k+1)4(k+1)(k+1)
limk|x2kx24k4(k+1)x2k4kk|=limk|x2kx24k4(k+1)×4kkx2k|
=limk|x24|kk+1|
=|x24|limk|kk(1+1k)|
=|x24|limk|11+1k|
=|x24

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Jeffrey Jordon
Answered 2021-12-17 Author has 2064 answers

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