For those who'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's easier to see what's happening.
I'd like to know what function this converges to:<p>rect(x) • rect(x/2) • rect(x/4) • ...<p>Where • is the convolution operator.<p>Unlike the series in the video, 1 + 1/2 + 1/4 + ... 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://en.m.wikipedia.org/wiki/Bump_function" rel="nofollow">https://en.m.wikipedia.org/wiki/Bump_function</a>
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's so much mathematical beauty out there to be discovered and admired.<p>But I guess money's better this way.