Question

How do we find the radius of convergence?

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.

Answer 1

`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`