Taylor series

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The Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval (a-r, a+r) is the power series

f(n)(a)
   ∑  ------  (x - a)n
   n=0  n!

Here, n! is the factorial of n and f (n)(a) denotes the n-th derivative of f at the point a.

If this series converges for every x in the interval (a-r, a+r) and the sum is equal to f(x), then the function f(x) is called analytic. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem.

If a = 0, the series is also called a Maclaurin series.

The importance of such a power series representation is threefold. First, differentiation and integration of power series can be performed term by term and is hence particularly easy. Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available. Third, the (truncated) series can be used to compute function values approximately.

Note that there are examples of infinitely often differentiable functions f(x) whose Taylor series converge but are not equal to f(x).

Several important Taylor series expansions follow.

xn
  ex  =  ∑  ----    for all x (see exponential function)
         n=0  n!
             ∞  (-1)n+1      
 ln(1+x)  =  ∑  ------  xn   for |x| < 1 (see natural logarithm)
             n=1   n
   1       ∞
 -----  =  ∑  xn     for |x| < 1 (see geometric series)
 (1-x)     n=0
            ∞
 (1+x)α  =  ∑  C(α,n) xn     for |x| < 1  and  all complex α (see binomial theorem and binomial coefficient)
            n=0
            ∞   (-1)n
 sin(x)  =  ∑  -------  x2n+1   for all x (see trigonometric functions)
            n=0 (2n+1)!
            ∞  (-1)n
 cos(x)  =  ∑  -----  x2n   for all x (see trigonometric functions)
            n=0 (2n)!
            ∞  B2n (-4)n(1-4n)
 tan(x)  =  ∑  --------------  x2n-1   for |x| < π/2 (see trigonometric functions)
            n=1     (2n)!
            ∞   E2n
 sec(x)  =  ∑  -----  x2n   for |x| < π/2 (see trigonometric functions)
            n=0 (2n)!
               ∞      (2n)!
 arcsin(x)  =  ∑  --------------  x2n+1   for all x (see trigonometric functions)
               n=0  4n(n!)2(2n+1)
               ∞   (-1)n
 arctan(x)  =  ∑  -------  x2n+1   for all x (see trigonometric functions)
               n=0 (2n+1)
             ∞     1
 sinh(x)  =  ∑  -------  x2n+1   for all x (see hyperbolic functions)
             n=0 (2n+1)!
             ∞    1
 cosh(x)  =  ∑  -----  x2n   for all x (see hyperbolic functions)
             n=0 (2n)!
             ∞  B2n 4n(4n-1)
 tanh(x)  =  ∑  -----------  x2n-1   for |x| < π/2 (see hyperbolic functions)
             n=1    (2n)!


               ∞    (-1)n (2n)!
 arsinh(x)  =  ∑  --------------  x2n+1   for |x| < 1 (see hyperbolic functions)
               n=0  4n(n!)2(2n+1)
               ∞    1
 artanh(x)  =  ∑  ------  x2n+1   for all x (see hyperbolic functions)
               n=0 (2n+1)

Here, the numbers Bk appearing in the expansions of tan(x) and tanh(x) are the Bernoulli numbers. The C(α,n) in the binomial expansion are the binomial coefficients. The Ek in the expansion of sec(x) are Euler numbers.

All these expansions are also valid for complex arguments x.