1 power series

\sum_{n=0}^\infty a_n(x - x_0)^n

for more, see semester 2

1.1 extended ratio/root test

l = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| or l = \lim_{n \to \infty} \sqrt[n]{|a_n|}

(Note: here l \ne l)

then:

R = \begin{cases} \frac{1}{l} & \text{if}\ l > 0 \\ \infty & \text{if}\ l = 0 \\ 0 & \text{if}\ l = \infty \\ \end{cases}