>>11035169Let's say our function doesn't converge for all "x", and we want to generate a series that will converge for a particular section of our curve. We can shift that section of the curve over to x=0 and then do our series. So like this:
And this time instead of plugging in x=0, we'll be plugging in x=a to zero out all the other terms. So follow all the same steps as before, you get this:
So like for ln(x+1), the taylor series only converges for . So if we want to calculate something further out, let's say ln(10), you need to do this sort of thing. You can do it as well for and other functions that converge everywhere, but you don't need to.
Often you can use the taylor series approximation to stand in for your function if you're doing things to it, since it calculates to your function. Or if you're doing something and you want to approximate to make you life easier, you can just take the first few terms of the taylor series. It will get you an approximation.
So like, sometimes in differential equations we say , or . Because we can't solve some of these differential equations with a sine and cosine function, so we simplify them down to get an approximation. And we can figure out how accurate our approximation is as well, often times it's pretty damn accurate for small angles, like 99.4%.
Using taylor series is absolutely fucking essential in the world of science if you're actually solving real problems.
