I was watching the 2023 MIT Integration Bee on YouTube, and for fun, I tried to evaluate the final problem myself, which is the following definite integral:
$$\int_0^1\left(\sum_{n=1}^\infty \frac{\lfloor 2^nx\rfloor}{3^n}\right)^2dx$$
I solved the infinite geometric series in the parenthesis and got $2x$, then squared it and integrated. My answer was $\frac{4}{3}$, which differed from the video's answer of $\frac{27}{32}$. I plugged the intergral into Wolfram Alpha, and got my answer. What is the flaw with my (and Wolfram Alpha's) solution, and how would one derive the correct answer? You can see a little bit of the contestant's work in the video, but it's hard to make out their process.
