# Solving for the Interval of Convergence

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Question: Find the open interval of convergence and test the endpoints for absolute and conditional convergence:

(x-2)^n / (2^n)(n^2)

Â© BrainMass Inc. brainmass.com December 24, 2021, 4:48 pm ad1c9bdddfhttps://brainmass.com/math/calculus-and-analysis/solving-interval-convergence-8889

#### Solution Preview

Solution. Let Fn(x)=(x-2)^n/{(2^n)(n^2)}. Let y=(x-2)/2, then

Fn(x)=Gn(y)=y^n/n^2. Consider series

G1(y)+G2(y)+...+Gn(y)+... (1)

The coefficient of the nth term is a(n)=1/n^2. Obviously, the limit L of a(n+1)/a(n) is 1 as n tends to ...

#### Solution Summary

In a step-wise response of about 145 words, this solution illustrates how an open interval of convergence is found. All calculations required are included.

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