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I got the following sum with which I want to prove one limit fact: $$ f_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} (a^t)^{n-t} $$ I want to prove that $f_n(a) \to 1$ while $n \to \infty$ for $a\...

Is the following true? Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$ $$ \beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}...