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anonymous
 5 years ago
Find the interval of convergence for the given power series sum (((x3)^n)/(n(5)^n)), n=1 to infinity.
The series is convergent
from x=______, left end included (Y,n):_____
to x=______, right end included (Y,n):____
anonymous
 5 years ago
Find the interval of convergence for the given power series sum (((x3)^n)/(n(5)^n)), n=1 to infinity. The series is convergent from x=______, left end included (Y,n):_____ to x=______, right end included (Y,n):____

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0By the Cauchy–Hadamard theorem a power series of the form \[\sum\limits_{n=0}^\infty a_n z^n\] has radius of convergence \[R=\frac{1}{\limsup\limits_{n\to\infty}(\sqrt[n]{a_n})}.\] In this case \[a_n=\frac{1}{n\cdot(5)^n}\] so \[\sqrt[n]{a_n}=\frac{1}{5\cdot\sqrt[n]{n}}\to \frac{1}{5} \quad(n\to\infty).\] Therefore R=5 and the series is convergent in the interval \[x\in(2,8).\] At x=2 you'll get the harmonic series which is divergent while in x=8 it's an alternating series converge to log(2). So the series is convergent in\[x\in(2,8].\]
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