I tried plugging bigger and bigger $n$'s into my calculator and the result obviously approaches $0$ (albeit oscillating between positive and negative).
So how do you prove that:
$$\lim_{n \to \infty} \left(-\frac{1}{2}\right)^n = 0.$$
algebraically?
I tried a few approaches, but nothing that follows the rules I learnt about limits.
Best Answer
You can use the squeeze theorem:
$$ -|1/2|^{x}\leq (-1/2)^{x}\leq|1/2|^{x}. $$
And, as you noted, both terms in the left and right of the above inequality go to $0$ as $x\to\infty$.