Exploring the generalized binomial theorem with my younger son

My younger son is reading Conway and Guy’s The Book of Numbers right now and one of the early sections in the book on the binomial theorem caught his attention. We talked about the binomial theorem for a bit and then I showed him a few examples of the generalized binomial theorem and he was really interested.

Today we talked about the ideas a bit more, starting with a reminder of what the theorem says:



Now we used the ideas from the last video to take a look at an approximation to \sqrt{2}.



Next we looked at an approximation that turns out to be one step more complicated – \sqrt{3}




Finally, we went back to Mathematica to look at how good the approximation from the last video was:



It was really fun showing my son some of these advanced ideas. I’m excited to explore these ideas with a few more approximations tomorrow.

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