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):____

Question

anonymous · Answer

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$$x\in(-2,8].$$