Ratio test takes the infinite limit of: which simplifies to: . The series diverges because the infinite limit is greater than one.
This series section is really bugging me. Am I allowed to use the word friggin? I can't get an answer that makes sense for n!/100^n with the Riemann symbol in front. I try to use the Ratio Test and then realize that the terms are positive so I don't need an absolute value.
I do some algebraic manipulation and come up with ((n+1)/(101)^n)) I don't see how I can take a limit/simplify the denominator here. Do I divide everything by ((101)^n))? Or did I do something wrong along the way and I should be getting a different result where I can see if it is <1 (converges) >1 (diverges) or = 1 (inconclusive).
As an aside, and this doesn't necessarily constitute another problem (hence no new thread). When I'm given a series n/(ln n)^n, can I use the rules of ln in the denominator and get n/((n)(ln n)) so the n's cancel out and the rest is easy?
I don't see how my method would be breaking any rules in math. I just want to check if I'm not mistaken.
I haven't gotten to logarithms yet as my teacher requested that I review algebra. I don't see how you got the n+1 in the numerator. There needs to be a 1 added to the numerator when the n comes down from being an exponent to multiply the denominator?
Stirling's formula tells us that (it does tell us what is but that is unimportant here):
,
so:
,
and eventually dominates for any so the terms diverge, and so the series itself diverges.
(that is for large enough and so the terms are bounded below by a multiple of a geometric series with common ratio greater than 1)
RonL