How do we find the radius of convergence?

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I know that we have to use the ratio test to find the

lim_(x->∞)Abs(U_(k+1)/U_k)

and then make a conclusion on based on what we get. However, I'm having trouble evaluating the same and making a conclusion.

Please correct me if I'm wrong somewhere.

1 Answer
May 2, 2018

R=oo

Explanation:

Apply the ratio test to the series:

sum_(k=0)^oo (k^3x^k)/(k!)

Evaluate the ratio:

abs(a_(k+1)/a_k) = abs ( ( ((k+1)^3x^(k+1))/((k+1)!) ) / ( (k^3x^k)/(k!) ))

abs(a_(k+1)/a_k) = abs ( (k+1)^3/k^3 (k!)/((k+1)!) x^(k+1)/x^k)

abs(a_(k+1)/a_k) = (k+1)^3/k^3 absx/(k+1)

abs(a_(k+1)/a_k) = (k+1)^2/k^3 absx

Now:

lim_(k->oo) (k+1)^2/k^3 = 0

so, whatever is the value of x, we have:

lim_(k->oo) abs(a_(k+1)/a_k) = 0

and the series is absolutely convergent.

We can conclude that the series converges for every x in RR and then the radius of convergence is R=oo