- #1
goraemon
- 67
- 4
Homework Statement
Does the following series converge or diverge? If it converges, does it converge absolutely or conditionally?
[itex]\sum^{\infty}_{1}(-1)^{n+1}*(1-n^{1/n})[/itex]
Homework Equations
Alternating series test
The Attempt at a Solution
I started out by taking the limit of ##a_n: Lim_{n\rightarrow\infty}(1-n^{1/n})##=1-1=0.
So via the alternating series test, the original series converges.
Next, I have to figure out whether it converges absolutely or conditionally, and this is where I'm stuck. I suppose I have to first find out whether the term ##a_n##, which is ##(1-n^{1/n})##, diverges or converges. But what test do I use for this? The limit test is silent because as I found above, the limit is zero. I've tried ratio test and limit comparison test to no avail. Root test doesn't seem to lead anywhere. It's not a geometric or telescoping series so those options are out. Maybe I could use comparison test, but I don't know what term I would use for comparison. Integral test seems really difficult. So I'm stuck and would appreciate any helpful pointers.