geometric series

Rationals Dense but Sparse

Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the rationals are dense in the real line… what this means is that any open set will contain some rational. So they are “everywhere” in the line, aren’t...

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Taylor-made Pi

After learning about the Taylor series for 1/(1+x) in calculus, you can find an interesting expression for Pi very easily. Start with 1/(1+w) = 1 – w + w2 – w3 + … Now substitute x2 for w: 1/(1+x2) = 1 – x2 + x4 – x6 + … Then integrate both sides...

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Cantor Set

Start with the interval [0,1]. Remove the (open) middle third of it, i.e. get (1/3, 2/3). Now remove the middle thirds of each of the remaining intervals, i.e. get (1/9, 2/9) and (7/9, 8/9). Continue this process ad infinitum. The points left over form a...

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