Hi, I need help with this series:
from n=2 to infinity, n/[ln(n)]^n
inf
Σ ____n_____
n=2 [ln(n)]^n
I tried the ratio test and when I went to take L'Hopital's I got into a big mess, can someone show me the solution?
Thank-you!
Okay thanks!
I went to a TA a last week and he confused me completely (had no idea what he was doing.. I ended up explaining the material to him). I can see now how it works, I forgot about my limit laws and was wondering what the heck to do with an indeterminate type of ∞^0/∞. This also was what got me into a huge mess with the ratio test...
You can use the ratio test
Expand the n+1 exponent
Divide stuff so that using limit laws so it is easier to take the limit
The first limit goes to 1 and the third limit goes to zero, but the second needs more work so raise the whole thing to the shared exponent
you can move the limit inside because of the power limit law
now you can use l'hospital's rule on the inside
with inverting and flipping you get this...
the limit goes to 1 and multiplied by zero gives zero which is less than 1 so by the ratio test...
converges
whoo, that was a mouthful
If, rather than applying a "black box" convergence test, you would like to "feel" why this series converges, you can note that as soon as , hence . So you see that the general term converges to 0 very quickly. Since for large (in fact, for all ), we have for large . Since the geometric series is well known to converge, your series converges.