anonymous
  • anonymous
Find the interval of convergence for the given power series sum (((x-3)^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):____
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
By 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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