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?