Power Series: Find Interval & Radius of Convergence

[Sentience]

Homework Statement

\Sigma (from index k = 1 until infinity)

Within the Sigma is the series : (k! * (x^k))

Homework Equations

Ratio Test : lim as k approaches infinity |a_(k+1) / a_k|

The Attempt at a Solution

When I apply the ration test to the series and simplify I get lim k --> inf (x * (k+1))

My confusion lies in the fact that the text answers this limit as infinity(which is pretty obvious if you're only taking k into consideration) and they pull the x out in front of the limit. I thought you could only do that with constants?

 

[HallsofIvy]

No, you can take out anything that does not depend on k. Whether it is a function of other things does not matter. The point is that you are checking convertence "pointwise"- that means that you are taking the limit as k goes to infinity for a fixed x. For the purposes of this problem, "x" is a constant.