For any $m\in\mathbb N$, let $S(m)$ be the digit sum of $m$ in the decimal system. 

For example, $S(1234)=1+2+3+4=10, S(2^5)=S(32)=5$. 

**Question 1** :Is the following true?
$$\lim_{n\to\infty}S(3^n)=\infty.$$

**Question 2** :How about $S(m^n)$ for $m\ge 4$ except some trivial cases?

**Remark** : This question has been asked previously on math.SE without receiving any complete answers, where spin proved that $\lim_{n\to\infty} \sup S(m^n )=\infty$ when $m$ is not a power of $10$. 

https://math.stackexchange.com/questions/501019/letting-sm-be-the-digit-sum-of-m-then-lim-n-to-inftys3n-infty



**Motivation** : I've got the following : 
$$\lim_{n\to\infty}S(2^n)=\infty.$$

**Proof** : The point of this proof is that there exists a non-zero number between the ${m+1}^{th}$ digit and ${4m}^{th}$ digit.

If 
$$2^n=A\cdot{10}^{4m}+B, B\lt {10}^m, 0\lt A,$$
then $2^n\ge {10}^{4m}\gt 2^{4m}$ leads $n\gt 4m$.
Hence, the left side can be divided by $2^{4m}$. Also, $B$ must be divided by $2^{4m}$ because ${10}^{4m}=2^{4m}\cdot 5^{4m}$. However, since
$$B\lt {10}^m\lt {16}^m=2^{4m},$$
$B$ can not be divided by $2^{4m}$ if $B\not=0$. If $B=0$, then the right side can be divided by $5$ but the left side cannot be divided by $5$. Hence, we now know that there is a non-zero number between the ${m+1}^{th}$ digit and ${4m}^{th}$ digit. Since $2^n$ cannot be divided by $5$, the first digit is not $0$. There exists non-zero number between the second digit and the fourth digit. Again, there exists non-zero number between $5^{th}$ digit and ${16}^{th}$ digit. By the same argument as above, if $2^n$ has more than $4^k$ digits, then $S(n)\ge {k+1}$. Hence,
$$n\log {2}\ge 4^k-1\ \ \Rightarrow \ \ S(n)\ge k+1.$$
Now we know that 
$$\lim_{n\to\infty}S(2^n)=\infty$$
as desired. Now the proof is completed.

However, I've been facing difficulty for the $m=3$ case. I've got $\lim\sup S(3^n)=\infty$.

**Proof** : Suppose that $3^n$ has $m$ digits. Letting $l=\varphi({10}^m)+n$, then 
$$3^l-3^n=3^n(3^{\varphi({10}^m)}-1).$$
Since this can be divided by ${10}^m$, we know that the last $m$ digits of $3^l$ are equal to those of $3^n$. Hence, we get $\lim\sup S(3^n)=\infty$.

However, I can't get $\lim_{n\to\infty}\inf S(3^n)$. Can anyone help?