Mathematics

This problem hinges on integration by parts combined with a power series expansion. Once you integrate $x^2 \log(1-x^2)$ by parts, you'll get a result involving $\log(1-x^2)$; then expand that logarithm as a series using $\log(1-u) = -\sum \frac{u^n}{n}$. The key insight is that evaluating your antiderivative at specific bounds will produce a series whose terms match the pattern in the sum you need to prove.

This problem hinges on integration by parts combined with a power series expansion. Once you integrate $x^2 \log(1-x^2)$ by parts, you'll get a result involving $\log(1-x^2)$; then expand that logarithm as a series using $\log(1-u) = -\sum \frac{u^n}{n}$. The key insight is that evaluating your antiderivative at specific bounds will produce a series whose terms match the pattern in the sum you need to prove.