Yes, and under weaker conditions.

Set $s = (1-e^x) > 0$, and multiply your partial sums by $s$; this has no
effect on log concavity. Then the resulting sequence is the Hadamard
product of the coefficients of $(\sum_{i=1}^n a_i)$ with $(s^n)$, each of
which is log concave (the first is the convolution of $(1,1,1,\dots,)$
with $(a_k)$), so the outcome is.

This argument requires only that $(a_k)$ be log concave (no monotonicity,
etc). [There is a quick reduction to $(a_k)$ being a finite sequence if
you don't like the possibly infinite convolution, since the $n$th partial
sum only involve the first $n$ terms of $(a_k)$, and log concavity is
checked just using the $n-1,n+1,n$th terms for each $n$.]