- #1
horsecandy911
- 17
- 0
Homework Statement
I have been straining to find convergence or divergence of a few infinite series. I have tried everything I can think of; ratio test, root test, trying to find a good series for comparison, etc. Here are the formulas for the terms:
#1
1
-------------
(ln(n))^ln(n)
#2
nth root(2) - 1
#3
(k*ln(k))
---------
(k+1)^3
Homework Equations
The Attempt at a Solution
For #1, I noticed that it is equivalent to:
(1/ln(n))^ln(n)
That is, we can raise the 1 on top to the ln(n) power also. Since 1/ln(n) < 1, this sort of resembles a convergent geometric series, but I am unsure how to prove convergence
For #2, the limit of terms approaches 0, so nth-term divergence test does not help us. I tried the ratio test but couldn't evaluate the limit of the resulting ratio; same for the root test. Tried Limit Comparison Test with nth root(2) for my second series, but got that the limit of the ratio was 0, which is inconclusive. I can't tell whether it diverges or converges.
For #3, I think that lnk<k^.5 for large k and (k+1)^3 is greater than k^3, so the terms are less than those of k^1.5/k^3 which = 1/k^1.5, which is a convergent p-series. So by the comparison test this would converge, but I am not sure whether I can use that ln(k)<k^.5 or even if its true.
Thanks for reading and helping!