Mike S.

asked • 08/13/17

Find the interval of convergence of the power series

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is a finite set, enter your answer using set notation.)

∑  (2n!)(x/3)^n
n=0

Andy C.

Perhaps maybe a comparison test.
You will need to find another series (perhaps in terms of (x/3)^n) that is  greater than (2n!)(x/3)^n which
converges. Converges follows by Sandwich Thoerem.
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08/14/17

2 Answers By Expert Tutors

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Dom V. answered • 08/23/17

Tutor
5.0 (119)

Cornell Engineering grad specializing in advanced math subjects

Andy C. answered • 08/13/17

Tutor
4.9 (27)

Math/Physics Tutor

Mike S.

It doesn't diverge. You are trying to find interval of convergence of the power series
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08/13/17

Andy C.

We are both partially correct.
 
The ratio test says that if the limit does not exist or is equal to one, then the test is inconclusive.
The series may converge or it may diverge. There are series that converge and diverge that
satisfy lim (An+1/An) = 1 or infinity.
 
You must first prove the series converges. Otherwise the radius of convergence is out of the question.
The strategy here was to get the limit in terms of x, then set it less than 1 to get the radius of convergence.
 
Find a convergence test that will work, and find the radius of convergence using that strategy.
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08/14/17

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