14. Calculate the product $p=1 \cdot 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot 16^{\frac{1}{16}} \cdot \ldots$

In № $15-20$ find the sums:
14. Let's write all powers as powers with base 2

$$
p=1 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{2}{4}} \cdot 2^{\frac{3}{8}} \cdot 2^{\frac{4}{16}} \ldots=2^{\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdots}
$$

The problem has been reduced to finding the sum $S=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\ldots$

Divide each term of this equality by 2 and subtract the obtained result from the first equality

$$
\frac{1}{2} S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots ; \quad \frac{1}{2} S=1, \quad S=2, \quad p=4
$$

86

15, 16, 17. $S=\frac{n}{n+1}, \quad S=\frac{n}{3 n+1}, S=\frac{n}{2 n+1}$ respectively. The specified sums can be found by constructing sequences of partial sums or by decomposing the terms of these sums into simpler fractions.