List of basic formulas related to power series:

  1. 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.
  2. 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.
  3. 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.
  1. 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 ).
  2. 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
    ]

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