Let f(x) = 1/((x-1)^(2)). Given an integer n, compute the nth derivative of f as f^((n))(x) = (-1)^(n)(n+1)!/((x-1)^(n+2)), which lets us write f as the Taylor series about x=0 whose nth coefficient is f^((n))(0)/n! = (-1)^(-2)(n+1)!/n! = n+1. We now compute the nth coefficient with a simple recursion. To show this process works, we make an inductive argument: the 0th coefficient is f(0) = 1, and the nth coefficient is (f(x) - (1 + 2x + 3x^(2) + ... + nx^(n-1)))/x^(n) evaluated at x=0. Note that each coefficient appearing in the previous expression is an integer between 0 and n, so by inductive hypothesis we can represent it by incrementing 0 repeatedly. Unfortunately, the expression we've written isn't well-defined at x=0 since we can't divide by 0, but as we'd expect, the limit as x->0 is defined and equal to n+1 (exercise: prove this). To compute the limit, we can evaluate at a sufficiently small value of x and argue by monotonicity or squeezing that n+1 is the nearest integer. (exercise: determine an upper bound for |x| that makes this argument work and fill in the details). Finally, evaluate our expression at the appropriate value of x for each k from 1 to n, using each result to compute the next, until we are able to write each coefficient. Evaluate one more time and conclude by rounding to the value of n+1. This increments n.
Sure, throw people in jail who haven't committed a crime, that'll fix all kinds of systemic issues
It could still be rust. Code is always the easy part. Design and organization and funding are hard
Sets aren't just for databases
Pikaaaaa
You shouldn't need to disable display names just to combat the 0.01% of people with annoying names. NFKC normalization is a better solution.
Depressing article. Freshly reinforced legal deterrent for anyone who might suggest that bicyclists are actually intended to use the roads. Can't say that or you might be liable.
If you've ever had a large coffee, it's like that. If you've ever had 3 large coffees and a heart condition, the same principle applies.
Sure, and VSCode without any plugins is a text editor, not an IDE.
You misread the comment
The ratio of consecutive terms of the Fibonacci sequence is approximately the golden ratio phi = ~1.618. This approximation gets more accurate as the sequence advances. One mile is ~1.609km. So technically for large enough numbers of miles, you will be off by about half a percent.
kogasa
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If "AI art is kinda bad" is part of the argument against AI art, it will only be used against us when AI art isn't that bad anymore.