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Power Series; Sum of Series; Estimate Using Terms; Convergence and Divergence (20 Problems)

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Please assist me with the attached problems, including:

8.7 Find the convergence set for the power series ...

8.8 Given the series (a) estimate the sum of the series by taking the sume of the first four terms. How accurate is the estimate? (b) How many terms of the series are necessary to estimate its sume with three-place accuaracy. What is this estimate?

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Twenty problems involving Power Series; Sum of Series; Estimate Using Terms; Convergence and Divergence are solved. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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Mathreview10a.

Find the convergence set for the following power series:

Look at these with the ratio test =

= 1 for n large The radius of convergence, R is 1 and the domain of convergence is -1<x<1

= 3 for n large The radius of convergence, R is 3 and the domain of convergence is -3<y<3 where y=x-2. The domain for x is then:
-1<x<5

= 1 for n large The radius of convergence, R is 1 and the domain of convergence is -1<y<1 where y=x-15. However, for this series starting at k=0 we have a problem as ln(0+1) is ln(1) is zero and we start off with a zero in the denominator. Thus for k=0 as the start the series diverges and makes little sense if k starts at one then the series converges for x in the domain 14<x<16.

= 1 for n large The radius of convergence, R is 1 and the domain of convergence is -3<y<3 where y=x-3. The domain for x is then: 2<x<4

for n large The radius of convergence, R is 2 and the domain of convergence is -2<y<2 where y=x + 3/2. The domain for x is then:
-(7/2)<x<(1/2)

let y = (-x) = 1 for n large The radius of convergence, R is 1 and the domain of convergence is -1<y<1 where y=-x. The domain ...

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