Does the following sum converge?

$newcommandan{lfloor a n rfloor}$ Let $a:=alphain(0,1)$. By induction on $m=0,1,dots$, $$sum_{k=0}^m binom nk(-1)^kBig(1-frac k{a n}Big) \ =(-1)^{m+1} (a+m-a n)frac{m+1}{an (n-1)},binom n{m+1}.$$ So, letting $S_n$ denote the sum in question, we have $$S_nsim(-1)^{lfloor a n rfloor+1}(a-{a n}) ,M_n,$$ where ${a n}$ is the fractional part of $a n$ and $$M_n:=frac1n,binom n{an+1}.$$ Let now $ntoinfty$. Depending on the arithmetical properties of $a$, the factor $(-1)^{lfloor a n rfloor+1}$ will alternate between $1$ and $-1$ and the factor $a-{na}$ will oscillate between $a-1<0$ and $a>0$, whereas $M_ntoinfty$, since eventually, for all large enough $n$, we have $binom n{an+1}gemin[binom n2,binom n{n-2}]=n(n-1)/2$. So, the sum $S_n$ will not converge to any limit.

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For an illustration, here are the connected graphs ${(n,c_a^n n^{3/2},S_n)colon n=1,dots,100}$ for $a=1/3$ (left) and $a=sqrt2-1$ (right), where $c_a:=a^a (1 - a)^{1 - a}in(0,1)$: