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**Ackbeet** Your basic idea works, but you want to clean up your notation. Mathematically, you're using the plain ol' comparison test: comparing $\displaystyle 2n^{n}$ with the sequence $\displaystyle n$. Once you've shown that $\displaystyle 2n^{n}>n$ for all $\displaystyle n>0$, then all you have to do is show that $\displaystyle (n)$ diverges. At that point, you invoke the comparison test, and you're done.

In particular, the statement $\displaystyle 2n^{n}>\infty$ is a bit sloppy. In addition, you can't write $\displaystyle n=\infty$; you can only take limits as $\displaystyle n\to\infty$.

Make sense?