What is Taylor series??

Question

What is Taylor series? Plz explain in simple words. I need the proof for the series as well. What's the difference b/w Taylor series and Maclaurin series? Any prerequisites needed to understand Taylor series better? Thanks a ton in advance!!

Answer 1

One should first look up what a Taylor series even is. Then, you shall find this:

$$\text{Taylor series:}\\f(x)=f(a)+f'(a)(x-a)+\frac12f''(a)(x-a)^2+\dots\\=\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n$$

And a Maclaurin series is simply the Taylor series at $a=0$. As far as notation goes, $f^{(n)}(a)$ is the $n$th derivative at $x=a$ and $n!=1\times2\times3\times\dots\times n$. A visual representation was always intuitive for me:

The black is $\ln(x+1)$ and the colored line is $\sum_{n=0}^N\frac{f^{(n)}(0)}{n!}x^n$.

Basically, you start with a point. Then you draw the tangent line. Then you draw... the quadratic line. And so on. I'm not quite sure what you mean by "proof", since you probably meant "derivation", since a function is equal to its Taylor series only if it is analytic...