Question

8 Σ n=1 (x-4)-1 3 n...


Calculate the interval and radius of convergence given (see image) Answer the following in order: 1. Limit expression for the ratio test: 2. Evaluation of convergence at the left and right endpoints(Interval of convergence: 3. Radius of convergence:
Transcribed: 18 Σ n = 1 η (x-4)-1 39

Answer

Given expression is

n=1(x-4)n-13nn

Let 

un=(x-4)n-13nn

then

un+1=(x-4)n3n+1(n+1)

Then

limnun+1un=limn(x-4)3n+1(n+1)×3nn(x-4)n-1=limnn(n+1)×(x-4)3=(x-4)3

For convergence

x-43<1=|x-4|<3       x<R, R is radius of convergence.=-3x-4<3=-3+4x-4+4<3+4=1x<7

So interval of convergence is [1,7)

and radius of convergence 3.

 

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