Partial sum of binomial coefficients

For some integer $z \ge 2$ and large integer $n$ and $t=\lceil \log n\rceil$, what is an approximate value for the following partial binomial sum? $$\sum_{i=0}^{n-t} \binom{n}{i}z^i .$$

Another related problem for the same parameters is approximating $$\sum_{i=n/t}^n \binom{n}{i}z^i .$$

Thanks.

• This is close to mathoverflow.net/questions/17202/… – Suvrit Nov 12 '15 at 5:32
• In both cases the sum is asymptotic to $(1+z)^n$ with great precision, since you have given almost all of the binomial summation. – Brendan McKay Nov 12 '15 at 5:38
• @BrendanMcKay true! I did not notice that $t$ is so small and took it to be arbitrary! – Suvrit Nov 12 '15 at 5:42
• @BrendanMcKay Thanks. Do you any reference or hint for its proof? – Aryo Z Nov 12 '15 at 6:13
• You can just be crude and say ${n \choose n-m} \leq n^m$ in this range. Thus the sum of the missing terms is at most $t n^t z^n$, which is exponentially smaller than $(1+z)^n$. – alpoge Nov 12 '15 at 7:18