Find limit for $n*C^n$ when $|C|<1$ Find $\lim_{n\to \infty}{n{C^n}}$ when: $|C|<1$
I want to use the squeeze theorem so I bounded it from below with: $C^n\to 0$
But I can't find the upper bound.

I know thats an old post but here is another solution
Bernoulli inequality states that:
$$\boxed{(1+\alpha)^n \geq n(1+n\alpha) \quad \forall r>-1, \text{ } n\in \mathbb{N^*}}$$

Given than $C<1$:
Let: $C = 1+\alpha \iff \alpha > -1$.
Also $n \in \mathbb{N} $

Therefore the contraints of Bernoulli inequality are satisfied and you can re-write your limit as:
$$ \lim_{n \to \infty} nC^n = \lim_{n \to \infty} n(1+\alpha)^n \geq \lim_{n \to \infty} n(1+n\alpha) = +\infty$$
Hence, by the squeeze theorem:
$$ nC^n  \to +\infty $$
