Evaluate: $3\cdot9^{\frac{1}{2}}\cdot27^{\frac{1}{4}}\cdot81^{\frac{1}{8}} \ldots$ 
Evaluate: $3\cdot9^{\frac{1}{2}}\cdot27^{\frac{1}{4}}\cdot81^{\frac{1}{8}} \dotsb$

Trial: Let
$$\begin{align}
P &= 3 \cdot 9^{\frac{1}{2}} \cdot 27^{\frac{1}{4}} \cdot 81^{\frac{1}{8}} \dotsb\\
\implies \ln P &=\ln3+\frac{1}{2} \ln 3^2+\frac{1}{2^2} \ln 3^3+\frac{1}{2^3} \ln 3^4 + \dotsb\\
&=\ln 3 \sum_{x=0}^{\infty}\frac{x+1}{2^x}
\end{align}$$
Then how I proceed. Is there any other simpler way to solve. Please help.

I think you are correct so far.  Use the facts that
$$\sum_{k=0}^{\infty} r^k = \frac{1}{1-r} $$
and
$$\sum_{k=0}^{\infty} k r^k = \frac{r}{(1-r)^2} $$
So you will have, with $r=1/2$:
$$\ln{P} = (2 + 2) \ln{3}  = 4 \ln{3} \implies P = 81 $$
