There is a well-known estimate for the sum of all binomial coefficients $\binom{n}{k}$ satisfying $k \leq \alpha n$ for some $\alpha$ satisfying $0 < \alpha \leq 1/2$:

$$ \sum_{k=0}^{\alpha n}\binom{n}{k} = 2^{(H(\alpha) + o(1))n}$$

where $H(\alpha) = -\alpha\log_2(\alpha) - (1-\alpha)\log_2(1-\alpha)$ is the binary entropy function.

My question is whether there exists a similar estimate when we weight the $k$-th binomial coefficient by $\lambda^k$ for some $\lambda > 0$. That is, I would like to estimate the following sum:

$$ \sum_{k=0}^{\alpha n} \binom{n}{k} \lambda^k $$