Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges. 1-2+4-8+...+(-1)^{n-1}2^{n-1}+...

allhvasstH

allhvasstH

Answered question

2020-12-03

Find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.
12+48+...+(1)n12n1+...

Answer & Explanation

i1ziZ

i1ziZ

Skilled2020-12-04Added 92 answers


Given:
The function given for the Taylor series is,
f(x)=(1x)1
a.The Taylor series for f(x)=(1x)1 centered at 0 is
11x=1+x+x2+x3+x4+...
n=0xn
Now, differentiate series with respect to x.ddx(1+x+x2+x3+x4+...)=1+2x+3x2+4x3+...
=n=1nxn1
b.To find the function represented by the differentiated series, differentiate f(x).
f(x)=ddx(11x)=1(1x)2
Therefore, the function represented by the differentiated series say h(x) is
h(x)=1(1x)2
c. Find the interval of convergence of the power series for the derivative.
The power series for the derivative is
f(x)=n=1nxn1
Use the Ratio test to compute the interval of convergence.
=limn|an+1||an|=limn|(n+1)xn||nxn1|
=limn(n+1n)|x|
=limn(n+(1+1n)n)|x|
=limn(1+1n)|x|
=(1+0)|x|
=|x|
The series converges for |x|<1.
Therefore, -1Now, we have to check the convergence of series at the endpoints of the interval.
At x=-1, the series becomes
n=1n(1)n1
As limnan=non exists
Therefore, by the series divergence test, the series diverges.
At x=1, the series becomes
n=1n(1)n1=n=1n
As limnan=
Therefore, by the series divergence test, the series diverges.
Hence, the interval of convergence is -1<x<1 or (-1, 1).
Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-17Added 2605 answers

Answer is given below (on video)

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