2. Which of the numbers is greater: $\underbrace{\sqrt{17 \sqrt{13 \sqrt{17 \sqrt{13 \sqrt{17 \ldots}}}}}}_{2018 \text { root signs }}$ or $17 \sqrt[3]{\frac{13}{17}}$?
Answer: the second.

Solution. Let $A$ be the first number, $B$ be the second. Then

![](https://cdn.mathpix.com/cropped/2024_05_06_84c54589ca5e85eb41a6g-1.jpg?height=85&width=1505&top_left_y=1334&top_left_x=281)

Since

$$
\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\ldots+\frac{1}{2^{2017}}<\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\ldots=\frac{1 / 2}{1-1 / 4}=\frac{2}{3}
$$

and

$$
\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\ldots+\frac{1}{2^{2018}}<\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\ldots=\frac{1 / 4}{1-1 / 4}=\frac{1}{3}
$$

the number $A$ is less than $17^{\frac{2}{3}} \cdot 13^{\frac{1}{3}}=B$.

Answer to option 2: the second. Answer to option 3: the first. Answer to option 4: the first.