Find the interval of convergence of the power series Σ [(2x)^n]/[(3n)!] from n=0 to infinity. Do I have to use the ratio test?
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Using the ratio test is generally a good idea for investigating convergence for almost most power series questions usually given.
r = lim(n→∞) |[(2x)^(n+1)/(3n+3)!] / [(2x)^n/(3n)!]
..= 2|x| * lim(n→∞) (3n)!/(3n+3)!
..= 2|x| * lim(n→∞) 1/[(3n+3)(3n+2)(3n+1)]
..= 0.
Since r = 0 < 1, this series converges for all x.
I hope this helps!
r = lim(n→∞) |[(2x)^(n+1)/(3n+3)!] / [(2x)^n/(3n)!]
..= 2|x| * lim(n→∞) (3n)!/(3n+3)!
..= 2|x| * lim(n→∞) 1/[(3n+3)(3n+2)(3n+1)]
..= 0.
Since r = 0 < 1, this series converges for all x.
I hope this helps!