Limit of the following term

Question

$$\lim_{n\to \infty}2^{n-k}(1-p)^n$$Where $n$ and $k$ are such that $0.4 < \frac{k}{n} < 1$, $0 < p < 0.5$ ?

I'm not sure how to deal with the relation between $k$ and $n$; I assume I have to substitute something in but I'm not sure, an help would be appreciated!

Answer 1

$$0.4 <\frac {k}{n }<1\implies 0 <1-\frac {k}{n}<0.6$$

$$\implies 1 <2^{1-\frac {k}{n}}<2^{0.6} $$

$$0 <p <0.5\implies 0.5 <1-p <1$$

thus

$$1<2^{n-k}<2^{0.6n}$$ and $$0.5^n<(1-p)^n <1$$ finally

$$0 <2^{n-k}(1-p)^n <2^{0.6n }$$

We can say nothing about the limit.

Answer 2

Note that $$0.4n<k<n\implies 2^{-n}<2^{-k}<2^{-0.4n}$$ Thus, $$(1-p)^n<2^{-k}2^n(1-p)^n<[2^{0.6}(1-p)]^n$$ We don't know if the limit exists or not.