Question on the ratio test for series convergence?

Question

So I recently answered a question on series convergence: http://answers.yahoo.com/question/index;_ylt=Avx5Ntvei6Ku672AVaviFpDAFQx.;_ylv=3?qid=20131206160406AAjecfW

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?

kb · Accepted Answer

Are you sure about this?

lim(n→∞) n^(n+1) / (n+1)^(n+1)
= lim(n→∞) 1 / [(n+1)^(n+1) / n^(n+1)]
= lim(n→∞) 1 / [(n+1)/n]^(n+1)
= lim(n→∞) 1 / (1 + 1/n)^(n+1)
= lim(n→∞) [1/(1 + 1/n)^n] * [1/(1 + 1/n)]
= 1/e * 1, using the limit definition of e with the first factor
= 1/e.

I hope this helps!

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Question on the ratio test for series convergence?

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