If you apply the ratio test, then the limit approaches 1 and the test is inconclusive. However, from an intuitive standpoint, this series should converge (see my comparison test) which shows that the exponents essentially differ by 2 and thus it should converge by the p-series test. But if you look at the result for the ratio test: you get the following limit:
lim_{x --> ∞} n^(n + 1) / (n + 1)^(n + 1)
This does indeed converge to 1 (the limit) and thus the ratio test is inconclusive. HOWEVER, it approaches 1 from the "bottom". That is, it's ALWAYS < 1 until you actually get to n = ∞. If you can show that the the limit approaches 1 from the bottom (always from the bottom) can this break the indeterminance of the ratio test? Or conversely if the limit approaches 1 from the top, can you say that the series definitely diverges?
更新1:
@kb - you are correct, the ratio test does work in this case. I incorrectly took (or really didn't take) the limit.
