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Math Help - ratio test to find the radius of convergence and interval

  1. #1
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    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
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  2. #2
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    Hello, gabet16941!

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

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

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


    The series converges for all values of x.

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

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