power’s out. hmmmm
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apparently won't be restored until tonight. taking this opportunity to do some math in the sun
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ran out of sun a few hours ago and couldn’t look at paper until i found a flashlight and hung it on my window
feeling very trad
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decided to do some trad math and currently attempting to check the computation of log 2 to three digits by hand
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still no power. computed log 2, log 3, and log 5 to three digits by hand. tomorrow i might learn how to churn butter idk
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they are several pages unfortunately. the main upshot is using the taylor series of hyperbolic arctangent to speed up convergence (it's a little more than twice as fast)
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so e.g. if you wanted to compute log 2 you could do it by computing it as log 1 / (1 - 1/2) = sum (1/2)^n / n, this converges but kinda slowly. there's a much funnier way to do it which is to compare the taylor series for log 3/2 = log 1 / (1 - 1/3) and log 4/3 = log (1 - 1/3)
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this gets you log 2 = log 3/2 + log 4/3 = 2 sum (1/3)^{2n-1} / (2n-1), whose terms grow like a power of 9 instead of like a power of 2 and so which converges more than 3 times as quickly (in the sense that you get more than 3 times as many digits of accuracy per term)
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whoops, that should've been log 4/3 = log (1 + 1/3), the point is that the even terms in the taylor series cancel when you add them and this means you're getting the taylor series of hyperbolic arctangent
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