# Thread: ratio test to find the radius of convergence and interval

1. ## ratio test to find the radius of convergence and interval

Use the ratio test to find the radius of convergence and interval of convergence of the power series.

i bealive it would be the sum from 0 to infinity of (3x)^n/n! and using the ratio test i get 3x but after that not sure how to get radius and the interval

2. Hello, gabet16941!

Use the ratio test to find the radius of convergence and interval of convergence.

. . $\displaystyle S \;=\;1 + 3x + \frac{9x^2}{2!} + \frac{27x^3}{3!} + \frac{81x^4}{4!} + \frac{243x^5}{5!} + \hdots$

$\displaystyle R \;=\;\frac{a_{n+1}}{a_n} \;=\;\frac{(3x)^{n+1}}{(n+1)!}\cdot\frac{n!}{(3x)^ n} \;=\;\frac{n!}{(n+1)!}\cdot\frac{(3x)^{n+1}}{(3x)^ n} \;=\;\frac{1}{n+1}\cdot3x$

Hence: .$\displaystyle \lim_{n\to\infty}R \;=\;\lim_{n\to\infty}\frac{3x}{n+1} \;=\;0$

The series converges for all values of $\displaystyle x.$

The interval of convergence is: .$\displaystyle (-\infty,\:\infty)$