My homework asks me to calculate (if it exists) the following limit:
$$\lim_{n\to\infty}{\frac{(1+(-1)^n)^n}{n}}$$
My thinking is: $(-1)^n$ would, as we all know, oscillate between 1 and -1, meaning that $(1+(-1)^n)$ would be either $0$ or $2$. Thus, for all odd cases: $$\lim_{n\to\infty}{\frac{0^n}{n}}=0$$ And then, for all even cases: $$\lim_{n\to\infty}{\frac{2^n}{n}}$$ Using Cauchy: $$\lim_{n\to\infty}{^n\sqrt{\frac{2^n}{n}}}$$ $$\lim_{n\to\infty}{\frac{^n\sqrt{2^n}}{^n\sqrt{n}}}$$ $$\lim_{n\to\infty}{\frac{2}{1}} = 2$$
And then, it follows that $$\lim_{n\to\infty}{\frac{2^n}{n}} = \infty$$ Which means that our original expression... has no limit?