1. The sequence $\log \sqrt{x}, \log \sqrt[4]{x}, \log \sqrt[8]{x}, \ldots$ is given. Determine the sum $S(x)$ of this sequence and sketch the graph of the function $S(x), x>0$.
Solution. The general term of the sequence is

$$
a_{n}(x)=\log \left(x^{\frac{1}{2^{n}}}\right)=\frac{1}{2^{n}} \log x
$$

therefore,

$$
\frac{a_{n}(x)}{a_{n-1}(x)}=\frac{\frac{1}{2^{n}} \log x}{\frac{1}{2^{n-1}} \log x}=\frac{1}{2}
$$

which means that the given sequence is geometric. If $\log x=0$, then $a_{n}(x)=0$. The sum of the corresponding series is

$$
S(x)=\frac{a_{0}(x)}{1-\frac{1}{2}}=2 \log x
$$

We leave it to the reader to sketch the graph of this function.