TITLE: Find all values of $k$ such that the series with terms $k^n / n^k$ converges.
QUESTION [1 upvotes]: Find all values of $k$ such that the series with terms $k^n / n^k$ converges.
I'm aware of some basic tests which I can apply (limit comparison test, direct comparison test, ratio test etc) but I don't see how any of them can help since I don't know what to compare it with and the ratio test doesn't work

REPLY [1 votes]: Nevermind, I applied the test wrong. It does work.
Consider the expression $k^{n+1}/(n+1)^k * n^k/k^n$ 
= $a_{n+1}/a_n$ 
= $k * n^k / (n+1)^k$ 
Now, in the limit as $n$ grows large, the fraction on the right approaches 1, so the whole thing approaches $k$. We need $|k| < 1$ for absolute convergence $<->$ conditional convergence in our case as all the terms are positive. 
The only other case is $k = 1$ in which the series becomes the harmonic series, which is known to diverge, or $k = -1$ which makes our series have an $n$ in the numerator, so it diverges. 
Thus, the series converges for $-1<k<1$
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