Example: Taylor expansion of the logarithm function

3.5.4 Example: Taylor expansion of the logarithm function

We will perform the Taylor expansion of the function \begin{eqnarray*} f(x)&=&\ln (1+ x) \quad \mbox{around} \quad x_0=0\\ f(0)& = & \ln (1) = 0\\ \frac{df}{dx}&=&\frac{1}{1+x} \quad \Rightarrow \quad \frac{df}{dx}(0) = 1\\ \frac{d^n f}{dx^n}&=& \frac{(-1)(-2)\ldots (-(n-1))}{(1+x)^n} \quad \Rightarrow \quad \frac{d^nf}{dx^n}(0) = (-1)^{n+1} (n-1)!\\ \Rightarrow f(x)&=& x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots = \sum_{n=1}^{\infty} (-1)^{n+1}\frac{x^n}{n}\end{eqnarray*}

This result we use to prove one fundamental equation of the exponential function \begin{eqnarray*} \ln \left( 1+\frac{x}{l}\right)^l & = & l \; \ln \left( 1+\frac{x}{l}\right) = l \sum_{n=1}^{\infty} (-1)^{n+1}\frac{x^n}{l^n\;n} \\ \Rightarrow \lim_{l \to \infty} \ln \left( 1+\frac{x}{l}\right)^l &=& x \\ \Rightarrow \lim_{l \to \infty} \left( 1+\frac{x}{l}\right)^l &=& \exp(x)\end{eqnarray*}

With frame

Taylor series and their application