Norm of a bounded linear functional

Question

could anyone please help me with the following question? Given $l^{1}:\rightarrow\mathbb{R}$ defined by $f(a_1,a_2,a_3,...,a_n,...)=\frac{2a_1}{3}+\frac{4a_2}{9}+\frac{8a_3}{27}+...+\frac{2^{n}a_n}{3^{n}}+...$ decide whether $f$ is a bounded linear functional. If so calculate $||f||$.

Answer 1

Look at $|f(a)|\leq |\frac{2a_1}{3}|+|\frac{4a_2}{9}|+...\leq |\frac{2a_1}{3}|+|\frac{2a_2}{3}|+...=\frac{2}{3}\|a\|_1$

And the case $a=(1,0,0,...)$.