TITLE: Does this sequence of ratios of digit sums have a limit?
QUESTION [6 upvotes]: I asked this question a few hours ago on MathStackExchange and there it received some attention but we still do not have a proof so I decided to ask it here also, in an unchanged form, and here it is:
Digit sums of numbers $3^m$ in base $10$ for $m=1,2,...,50$ are:
$3,9,9,9,9,18,18,18,27,27,27,18,27,45,36,27,27,45,36,45,27,45,54,54,63,63,81,72,72,63,81,63,72,99,81,81,90,90,81,90,99,90,108,90,99,108,126,117,108,144$.
Ratios $\dfrac {ds_{10}(3^m)}{ds_{10}(3^{m+1})}$ for $m=1,2,...,49$ to three decimal places are:
$0.333,1.000,1.000,1.000,0.500,1.000,1.000,0.666,1.000,1.000,1.500,0.666,0.600,1.250,1.333,1.000,0.600,1.250,0.800,1.666,0.600,0.833,1.000,0.857,1.000,0.777,1.125,1.000,1.142,0.777,1.285,0.875,0.727,1.222,1.000,0.900,1.000,1.111,0.900,0.909,1.100,0.833,1.200,0.909,0.916,0.857,1.076,1.083,0.750$

Does there exist limit of the sequence $a(m)=\dfrac {ds_{10}(3^m)}{ds_{10}(3^{m+1})}$?

I cannot resist to note some kind of chebyshevness of this question (if there is one) because we know that Chebyshev proved that if limit in the prime number theorem exists then it must be equal to $1$. It could be that this is also the case here.
I also welcome any computational effort and results obtained from such an experimental work if the proof is out of reach.

REPLY [3 votes]: Experimentally, the sequence converges to $1,$ at the logarithmic rate suggested in Fedor's answer. Here is the graph for the first 20000 numbers:

Now, when we fit the actual $ds_{10}(3^m),$ we get the following suggestive graph:

Whose slope is pretty close to Fedor's "probabilistic" value. HOWEVER convergence is slow - empirically, Fedor's $o(1)$ term is actually of order $1/\log m.$