The Taylor series about $x = 0$ of any given function:

$$\begin{aligned} \displaystyle f(x) = \sum_{k=0}^\infty f^{(k)}(0)\frac{x^k}{k!} \\ \end{aligned}$$

Some interesting observations:

• The long polynomial of $\displaystyle e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$ can easily be verified to be just the Taylor series about $x = 0$ of $e^x$
• The Taylor series about $x = 0$ of a polynomial is the polynomial per se
• Geometric series such as $1 + x + x^2 + x^3 + \cdots$ can easily be verified to be the Taylor series about $x = 0$ of $\displaystyle \frac{1}{1-x}$

# posted by rot13(Unafba Pune) @ 4:37 PM