Taylor Series For Expanding sinx , cosx, exp(x) 

A one-dimensional Taylor series is an expansion of a real or complex-valued polynomial or  function f(x) about a point x=a is given by

	=	factorial of n
	=	real or complex number
	=	nth derivative of f evaluated at the point a

Expanding

 f(x)= 
 

  If a=0, the expansion is known as a Maclaurin series.

      let  f(x) = e^x

            f'(x) = e^x

            f''(x) = e^x  

e^x = 

              

at a = 0 ,   

f’(0) = e0 =1  

f’’(0) = e0=1

f’’’(0) = e0 = 1

ex=1+x(1)+x22!(1)+x33!(1)+⋯
ex=1+x+x22!+x33!+x44!+⋯

xx

  

in similar fashion sinx and cosx can also derived from Taylor series

Similarly, one can derive for the hyperbolic sinh and cosh the expansions

sinhx= x+x33!+x55!+

coshx =  1+x22!+x44! + 

Let us derive for log(x) at x=1
f(x) = log x222log(x) 

=n=1∑∞​[n(−1)n+1​(x−1)n]