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How to evaluate the limit $$\lim_{n \to +\infty}(-1)^n\frac{n^n + \ln n}{\cos(n\pi)(n + \pi)^n}$$?
I was suspecting that the sequence does not converge, but the ratio test, which is the only one that I know, turned out to be useless. The limit of the ratio $\frac{a_{n+1}}{a_n}$ is $1$, so the test is inconclusive.
It turns out the result is $\boxed{\displaystyle e^{-\pi}}$, but I fail to see how one would reach it.
Any suggestions?
Observe that $\cos(\pi n)=(-1)^n$, hence $$ (-1)^n\frac{n^n+\log n}{\cos(\pi n)(n+\pi)^n}=\frac{n^n+\log n}{(n+\pi)^n}=\frac{1+\frac{\log n}{n^n}}{\left(1+\frac{\pi}{n}\right)^n} $$ Now the numerator goes to $1$, and the denominator is a notable limit, it goes to $e^{\pi}$, hence you get what you wanted.
