常见函数的泰勒级数展开.pdfVIP

Taylor Series for Functions of One Variable f ′′(a )(x −a )2 f (n−−1) (a)(x − a)n − 1 22.1. f (x ) = f (a ) + f ′(a )(x −a ) + + ⋅⋅⋅ + + Rn 2 ! (n − 1)! where R , the remainder after n terms, is given by either of the following forms: n (n ) (ξ )( )n f x −a 22.2. Lagrange’s form: Rn = n ! (n ) (ξ )( ξ )n−1 ( ) f x − x −a 22.3. Cauchy’s form: Rn = (n − 1)! The value x, which may be different in the two forms, lies between a and x . The result holds iff (x) has continuous derivatives of order n at least. If lim Rn = 0, the infinite series obtained is called the Taylor series for f (x) about x a. If a 0, the series n→∞ is often called a Maclaurin series. These series, often called power series, generally converge for all values of x in some interval called the interval of convergence and diverge for all x outside this interval. Some series contain the Bernoulli numbers B and the Euler numbers E defined in Chapter 23, pages n n 142143. Binomial Series n n n−1 n(n − 1) n− 2 2 n(n − 1)(n − 2) n− 3 3 22.4. (a + x ) = a + na x + a x + a x + ⋅⋅⋅ 2 ! 33! = an + ⎛n⎞ an−1x + ⎛n⎞ an− 2xx 2 + ⎛n⎞ an− 3 x 3 + ⋅⋅⋅ ⎜ 1 ⎟ ⎜ 2 ⎟ ⎜ 3 ⎟