The argument describes an algorithm that can be translated into code.
1/(1-x)^(2) at 0 is 1
(1/(1-x)^(2) - 1)/x = (1 - 1 + 2x - x^(2))/x = 2 - x at 0 is 2
(1/(1-x)^(2) - 1 - 2x)/x^(2) = ((1 - 1 + 2x - x^(2) - 2x + 4x^(2) - 2x^(3))/x^(2) = 3 - 2x at 0 is 3
and so on
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.
Start with the goal to create something, be it a console app, website, web api, or game. It's hard to just study a language abstractly and learn it. Use the Microsoft Learn documentation as reference, and look for open source .NET projects on GitHub to get different perspectives on how to build things with .NET. There is a free course on freecodecamp that will get you started by building an app, and I believe it was done in partnership with Microsoft
SQLite is one of the best tested codebases in existence. Having only so many variables per line means nothing
Don't think you can stack 2.4 million bananas on top of each other. By volume you'd need like 10^16 bananas to form everest
I'm a big fan of stuff like this. Not this specifically though.
You can also get a bluetooth amp/dac and plug your wired headphones into there. I use a Qudelix 5k for my IEMs at home and I can just put it in my pocket if I want to take them out.
Your first sentence asserts the claim to be proved. Actually it asserts something much stronger which is also false, as e.g. 0.101001000100001... is a non-repeating decimal which doesn't include "2". While pi is known to be irrational and transcendental, there is no known proof that it is normal or even disjunctive, and generally such proofs are hard to come by except for pathological numbers constructed specifically to be normal/disjunctive or not.
Impossible to say how the screenshot came about. At some point, the semantics of someone's data was misinterpreted. Something was called a hard drive when it isn't, something was treated as a capacity when it isn't
Everyone has their own way of doing things. In order to integrate them, you have to create your own way of doing things. Later, other people have to integrate your new schema with their own. It's just the nature of working with many different companies
I'm a hobbyist speed typer (200wpm+), generally prefer linear switches. I do bottom out almost always. To reduce the impact of bottoming out, if this is an issue for you, you can:
use a softer and/or more flexible plate. An aluminum or brass plate is very stiff and will absorb less of the impact compared to an FR4 or polycarbonate plate. The mounting style of the keyboard can also affect this, e.g. a gasket mount has the pcb "floating" on rubber pads that absorb shock, and a plate that is screwed directly into a metal chassis will absorb almost nothing. The plate/pcb can have flex cuts added to improve flexibility and absorb more shock.
use switch springs with a higher actuation force. Common choice is 63.5g or 68g, which is a little heavier than the Akko switches' ~45g. The spring can also have a variable profile such that the resistance increases more as the spring is depressed, so it kind of cushions the impact a tiny bit. I use extra long springs which has the opposite effect, the response curve is more constant.
use rubber o-rings on the switches. This will make them feel squishy and I don't really recommend it, but it's an option if replacing your keyboard isn't.
FWIW I mostly use an Odin75 keyboard with an FR4 plate and stock alpaca switches. This is gasket mount + soft plate with lots of flex cuts, so it's a reasonably soft typing experience.
If you take immortality, you also probably need to take healing. Being mortally wounded and unable to die sounds, uh, bad.