# College Math Teaching

## November 18, 2013

### And I get sloppy….divergence of n!x^n

Filed under: calculus, sequences, series — Tags: — collegemathteaching @ 9:29 pm

In class I was demonstrating the various open intervals of absolute convergence and gave the usual $\sum k!x^k$ as an example of a series that converges at $x = 0$ only. I mentioned that “$\sum k!x^k$ doesn’t even pass the divergence test”, which, as it turns out, is true. But why? (yes, it is easier to just use the ratio test and be done with it)

Well, I should have noted: if $x > 0$, then $x > \frac{1}{m}$ for some integer m, then for $k > m$ we have $k!x^k > \frac{1*2*3...*m *(m+1)*(m+2)...*k}{m*m*m...*m*m*m...*m}$ and one can see that this is a finite number times a number which is growing without bound. Hence the sequence of terms of the series grows without bound for any positive value of $x$.