List of basic formulas related to power series:
- The general form of a power series is:
[
f(x) = \sum_{n=0}^{\infty} a_n (x – c)^n
]
where ( a_n ) are the coefficients of the series, ( c ) is the center of the series. - Radius of convergence:
[
R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}
]
or
[
R = \lim_{n \to \infty} \frac{|a_n|}{|a_{n+1}|}
]
if the limit exists. - Convergence of the series:
- A series converges absolutely if ( \sum_{n=0}^{\infty} |a_n (x – c)^n| ) converges.
- A series converges conditionally if it converges but not absolutely.
- Taylor formula (for the function ( f(x) ) at the point ( c )):
[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x – c)^n
]
where ( f^{(n)}(c) ) is the n-th derivative of the function at the point ( c ). - Maclauren’s formula (a special case of Taylor’s formula when ( c = 0 )):
[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} x^n
]