cos(n*pi/n^2)
i have to prove that this sequence is converging to 0. i am unable to see how this would converge to 0. any ideas?
As you have it written $\displaystyle \lim_{n\to\infty}\cos\left(\frac{n\pi}{n^2}\right) = 1$ because as $\displaystyle n\to\infty, \frac{n\pi}{n^2}\to 0$ and $\displaystyle \cos(0)=1$.
Perhaps you meant $\displaystyle \lim_{n\to\infty}\frac{\cos(n\pi)}{n^2}$? This does equal zero because $\displaystyle \cos(n\pi)\leq 1$ for all $\displaystyle n$ so $\displaystyle \lim_{n\to\infty}\frac{\cos(n\pi)}{n^2} \leq \lim_{n\to\infty}\frac{1}{n^2} = 0$
And to add to that, now the "squeeze" part:
$\displaystyle -1\leq\cos(n\pi)\leq 1$ for all $\displaystyle n$ so $\displaystyle \lim_{n\to\infty}\frac{-1\ \ }{n^2} \leq\lim_{n\to\infty}\frac{\cos(n\pi)}{n^2} \leq \lim_{n\to\infty}\frac{1}{n^2} $
Now both "ends" converge to 0, squeezing the middle to zero.