We need to look at the ratio of successive terms, a_{n+1}/a_{n}, and check if in the limit nāā the ratio is > or < than 1:
((n+1)^{100}/100^{n+1})/(n^{100}/100^{n}) = ((n+1)^{100}/n^{100})/(100^{n+1}/100^{n}) = (1+1/n)^{100}/100
The limit is 1/100, which is less than 1, therefore by the ratio test, the series converges.
Conceptually: the exponential function in the denominator increases much faster than the power function in the numerator, so their ratio gets small fast.
10/23/2013

Andre W.