$ \fcolorbox{white}{yellow}{Power Series (expansion at center $x=0$): $\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots$ }$ |
$ \fcolorbox{white}{yellow}{or Power Series (expansion at center $x=a$): $\sum_{n=0}^\infty a_n (x-a)^n = a_0 + a_1 (x-a) + a_2 (x-a)^2 + a_3 (x-a)^3 + \cdots$ } $ |
$\fcolorbox{white}{yellow}{ Radius of convergence $R$: the series is convergent for $|x-a|< R$. }$ |
$\frac1R = \lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} \fcolorbox{white}{yellow}{Caution: the formula fails if the limit does not exist}$
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