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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 $a\in ]0,1[$ (if true. I plotted and it looks like it were true). As you see this is kind of binomial sum, but instead of multiplying by $a^{n-t}$ we are powering. I wrote the sum without sigma sign and tried to prove using members, however I stuck because the size of the sum still growing.

Any help would be appreciated.

Thank you.

[edit] If it is possible the speed of decreasing is also interesting, in terms of small-o.

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 $a\in [0,1)$ if true. (I plotted and it looks like it were true). As you see this is kind of binomial sum, but instead of multiplying by $a^{n-t}$ we are powering. I wrote the sum without sigma sign and tried to prove using members, however I stuck because the size of the sum still growing.

Any help would be appreciated.

Thank you.

[edit] If it is possible the speed of decreasing is also interesting, in terms of small-o.

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 $0 \geq a < 1$ (if true. I plotted and looks like it is true). As you see this is kind of binomial sum, but instead of multiplying by $a^{n-t}$ we are powering. I wrote the sum without sigma sign and tried to prove using members, however I stuck because the size of the sum still growing.

Any help would be appreciated.

Thank you.

[edit] If it is possible the speed of decreasing is also interesting, in terms of small-o.

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 $a\in ]0,1[$ (if true. I plotted and it looks like it were true). As you see this is kind of binomial sum, but instead of multiplying by $a^{n-t}$ we are powering. I wrote the sum without sigma sign and tried to prove using members, however I stuck because the size of the sum still growing.

Any help would be appreciated.

Thank you.

[edit] If it is possible the speed of decreasing is also interesting, in terms of small-o.

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 $0 < a < 1$ (if true. I plotted and looks like it is true). As you see this is kind of binomial sum, but instead of multiplying by $a^{n-t}$ we are powering. I wrote the sum without sigma sign and tried to prove using members, however I stuck because the size of the sum still growing.

Any help would be appreciated.

Thank you.

[edit] If it is possible the speed of decreasing is also interesting, in terms of small-o.

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 $0 \geq a < 1$ (if true. I plotted and looks like it is true). As you see this is kind of binomial sum, but instead of multiplying by $a^{n-t}$ we are powering. I wrote the sum without sigma sign and tried to prove using members, however I stuck because the size of the sum still growing.

Any help would be appreciated.

Thank you.

[edit] If it is possible the speed of decreasing is also interesting, in terms of small-o.