## anonymous 5 years ago 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):____

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