MathNet 2022-part-06 — 196 problems (EN–VI) Question 64

For a nonnegative integer $n$, let $s(n)$ be the sum of digits of the binary representation of $n$. Prove that
$$
\sum_{n=0}^{2^{2022}-1} \frac{(-1)^{s(n)}}{2022+n}\gt 0
$$