Lesson 2

Some standard expansions are given below. For each function, we state (without proof) the radius of convergence of the corresponding series.

• Exponential  (\(x \in \mathbb{R}\)): \[e^{x}=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\dotsb=\sum_{n=0}^{\infty}\frac{x^n}{n!}\ .\]


• Natural logarithm  (\(|x|<1\)): \[\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dotsb=\sum_{n=0}^{\infty}\frac{(-1)^n}{n+1}x^{n+1}\ .\]


• Binomial expansion  (\(|x|<1\)): \[(1+x)^{\alpha}=\sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n} \ ,\] where \[\binom{\alpha}{n}=\frac{\alpha(\alpha-1)\dotsb(\alpha-n+1)}{n!}\ .\]