Researchers thought this was a bug (Borwein integrals) [video]

For those who&#x27;d like to read a quick description of what this video is about:<p>It shows a sequence of integrals following a very simple pattern. The first seven integrals in the sequence all evaluate to pi. The eighth integral inexplicably evaluates to pi - 0.0000000000462... and from that point on the pattern deviates from pi.<p>The video goes on to explain how such a seemingly perfect pattern can suddenly break by relating this sequence of integrands to a simpler one where it&#x27;s easier to see what&#x27;s happening.

I&#x27;d like to know what function this converges to:<p>rect(x) • rect(x&#x2F;2) • rect(x&#x2F;4) • ...<p>Where • is the convolution operator.<p>Unlike the series in the video, 1 + 1&#x2F;2 + 1&#x2F;4 + ... converges. So this function has compact support, and the value at 0 does not dip.<p>I expect it to be a <a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Bump_function" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Bump_function</a>

I’m waiting for the convolution video he promises in the video and the comments.

I love this.<p>I sometimes regret not studying pure math in college, and going down the software engineer (<i>ahem</i>, code monkey) route. There&#x27;s so much mathematical beauty out there to be discovered and admired.<p>But I guess money&#x27;s better this way.

I love all the videos on this channel. The partial differential equations tour is incredible too.

Grant Sanderson redeems the entire Internet.