How do we find the radius of convergence?

Question

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.

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Answer 1 · 2018-05-02T13:06:30

#R=oo#

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#