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