It's not too bad, once you consider that everything in each term is a constant value, except for "x" itself.
So the numerator of each term is the product of three linear factors, like (x-4)(x-2)(x-6), which should produce a cubic, like x³ - 12x² + 44x - 48. Then the denominator of each term is a pure constant, so it would be like taking that cubic and dividing it by 4, getting 0.25x³ - 3x² + 11x - 12. Then the yₙ terms are also constants, so no different than doing something like multiplying by 2, getting you something like 0.5x³ - 6x² + 22x - 24, if I take that example a bit too far. And at that point it's just the sum of four cubics, which will be cubic as long as the x³ terms don't perfectly cancel out - which I believe would only happen if the four pairs of points used to make the function were all perfectly laying on the same line or parabola.
The construction's also pretty clever: OP said the point was to fit the function to the four points (x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄). Let's say we set x = x₂, then. Because (x-x₂) appears in the numerator of every term but the second term, every term but the second term will have a 0 in its numerator and cancel out - so we only need to consider the second term. Its numerator is then (x₂-x₁)(x₂-x₃)(x₂-x₄) - exactly the same as its denominator. So they both cancel out, leaving only y₂ - meaning we get P₃(x₂) = y₂, as desired.
Guess I'll just come out and say it. I'm a mixed fraction fan. 23+2/3 instantly tells you it's "23 and a bit", unlike 74/3, and it's more accurate than 23.67.
For me the problem is notation, putting a number in front of a fraction usually means multiplication and when giving a solution in anything but maths, the needed accuracy can be achieved with decimals
my favourite part is when you do by parts and it results in integral(say equation 1), then you do another and it results in original integral you started with(say equation 2).
so, you go clever and add both and find the result. it's clever and satisfying.
Except the first assumption that e^x = its own integral, everything else actually makes sense (except the DX are in the wrong powers). You simply treat the "1" and "integral dx" as operators, formally functions from R^R into R^R and "(0)" as calculating the value of the operator on a constant-valued function 0.
EDIT: the step 1/(1-integral) = the limit of a certain series is slightly dubious, but I believe it can be formally proven as well.
EDIT 2: I was proven wrong, read the comments
By far the most complicated part is the fact that the ratio of successive terms in the Fibonacci sequence approaches a specific number (which happens to be the golden ratio, which happens to be close to the ratio of km/mi).
What is the point of chili without meat? It would just be a bowl of beans and spices (or just a bowl of spices since beans really don't belong in chili).
Next time someone brings up Kansas in conversation (why would they), and if they imply nothing is there, you can contradict their statements easily. Kansas is invertible and therefore nonzero. QED.
Edit: Oh actually it's a function here never mind.
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