Dervation

Taylor Series

Keywords: taylor series, remainder, Lagrange

The Taylor series of a function f (x), that is infinitely differentiable at number a, is the power series:

Tx=n=0f(n)an!x-an

or in other form with (n-1) term and the remainder:

Tx=k=0n-1f(k)ak!x-ak+Rn

Lagrange's form

Rn=f(n)ξx-ann!

Cauchy's form

Rn=f(n)ξx-ξn-1x-an-1!

Taylor Serie of some elementary functions

Exponential and Logarithmic functions

ex=n=0xnn!  ; x
ln1+x=n=0-1nn+1xn+1  ; x<1

Geometric series

11-x=n=0xn  ; x<1
xm1-x=n=mxn  ; x<1

Binomial series

1+xα=n=0αnxn  ; x<1 α

Trigonometric functions

sin x=n=0-1n2n+1!x2n+1  ;  x
cos x=n=0-1n2n!x2n  ;  x

Hyperbolic functions

sh x=n=012n+1!x2n+1  ;  x
ch x=n=012n!x2n  ;  x