What's taylor series and when is it used?

Question

I am an eighth grader, and I just learned L'hopital's rule and taylor series. I perfectly understood the L'hopital's rule, but I really don't understand the taylor series. Can anybody explain me about it?

Answer 1

Taylor series are used to approximate certain functions with more than lines. Usually in calculus, we learn we can approximate a function, say $f(x)$, with at some point $x=a$ with a line provided that function is "nice" around that point. So, we can have

$$ f(x) \approx Ax + B $$

but this only works for value near $x=a$. If we get further away from that point, the approximation is not longer a good one. That is why Taylor Series are used. Basically, a taylor series is an infinite polynomial

$$ c_0 + c_1 (x-a) + c_2 (x-a)^2 + ... $$

that approximates some function $f(x)$ around given point $x=a$. The more terms we take from the series, the better the approximation.

Answer 2

A Taylor series is another way to write functions. Like $\sin x$ is a function and it's equivalent to $x - x^3/6 + x^5/5! - x^7/7! + \cdots $ which is the Taylor series for $\sin x$. Try going to desmos.com and graphing $\sin x$ and also graphing the series I wrote above and think about what you're seeing. A general understanding of sequences, series, and I believe the Mean Value Theorem are what you'll need if you want to know what Taylor series are at a deeper level. But at the end of the day they are exactly the function they describe (within certain domains). They are very useful for approximating functions since, you'll notice Taylor series is just a bunch of polynomials, which are easy to deal with.