2 clarify what is constant.

If . I'm taking $α<λ/(λ+1)$ (\lambda$and these are$\alpha$to be constants independent of which do not vary as$n$) n\to\infty$.

If $α<λ/(λ+1)$ then the sum is within a constant of the last term. In fact the largest terms are approximately in geometric progression so you can get it quite accurately by computing the ratio.

If $α>λ/(λ+1)$, almost all of the complete binomial expansion is present, so the sum equals $(1+o(1))(1+λ)^n$.

If $α≈p$, then Russell's normal approximation will be good. (This needs some work to clarify whether the geometric approximation of the lower tail is good right up to the point where the normal approximation begins to be good. I think it is.)

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If $α<λ/(λ+1)$ (and these are constants independent of $n$) then the sum is within a constant of the last term. In fact the largest terms are approximately in geometric progression so you can get it quite accurately by computing the ratio.
If $α>λ/(λ+1)$, almost all of the complete binomial expansion is present, so the sum equals $(1+o(1))(1+λ)^n$.
If $α≈p$, then Russell's normal approximation will be good.