Question 56
Let $f_{n} = 2^{2^{n}} + 1$, $n = 1, 2, 3, \ldots$, be the Fermat's numbers. Find the least real number $C$ such that
$$
\frac{1}{f_{1}} + \frac{2}{f_{2}} + \frac{2^{2}}{f_{3}} + \ldots + \frac{2^{n-1}}{f_{n}} \lt C
$$
for all positive integers $n$.
$$
\frac{1}{f_{1}} + \frac{2}{f_{2}} + \frac{2^{2}}{f_{3}} + \ldots + \frac{2^{n-1}}{f_{n}} \lt C
$$
for all positive integers $n$.