Let's Prove a Catalan's Constant Identity

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In this video we show that the integral from 0 to 1 of arctan(x)/x dx is equal to the sum from n=0 to infinity of (-1)^n/(2n+1)^2. We used this identity in my latest Math and Minecraft episode.

You can see that episode here:

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Nice one. Just a minor point for viewers to note that doesn't matter here but might in another derivation with different integration limits is the range of convergence for the 1/(1-(-t^2)) infinite series being |t^2|<1. In this case, with x going from 0 to 1, the upper limit of the inner integral is t=x<=1 which violates the range of convergence of the series when t can equal 1. This is trivially resolved in this case by letting the x integral go from 0 to q (with q<1) and then taking the limit of q->1 which makes it more rigorous (and complicated), but doesn't change the end result.

GandalfTheWise
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Fascinating video! Determining the precise values of those Catalan constants appears to be quite a challenging puzzle, doesn't it? Perhaps it's so intricate that we might contemplate giving up and saying, "Let's forget about it!" Using a placeholder like 'G' could potentially offer a more straightforward approach, given the difficulty in pinpointing the exact numerical values.

RSLT
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Nice I would have looked away from nested integrals thinking they would be way too complicated.

aahaanchawla
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I just expected "and thats the good place to stop" for a moment lol

kemalkayraergin
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Awesome! It's really interesting!

MathForLife
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Nice video, Im happy to see arctan instead of tan^(-1) xd

cookieman
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I have another way to verify this, we can use the series for arctan x and divide it by x, and solve for its integral from 0 and 1 and its just the series for the Catalan's constant!

Kdd