Constructing a Maclaurin Series
Key Questions

#f(x)=coshx=sum_{n=0}^infty{x^{2n}}/{(2n)!}# Let us look at some details.
We already know
#e^x=sum_{n=0}^infty x^n/{n!}# and
#e^{x}=sum_{n=0}^infty {(x)^n}/{n!}# ,so we have
#f(x)=coshx=1/2(e^x+e^{x})# #=1/2(sum_{n=0}^infty x^n/{n!}+sum_{n=0}^infty (x)^n/{n!})# #=1/2sum_{n=0}^infty( x^n/{n!}+(x)^n/{n!})# since terms are zero when
#n# is odd,#=1/2sum_{n=0}^infty{2x^{2n}}/{(2n)!}# by cancelling out
#2# 's,#=sum_{n=0}^infty{x^{2n}}/{(2n)!}# 
The MacLaurin series of a function
#f# is a power series of the form:#sum_(n=0)^(oo) a_n x^n# With the coefficients
#a_n# given by the relation#a_n=(f^((n))(0))/(n!),# where
#f^((n))(0)# is the#n# th derivative of#f(x)# evaluated at#x=0# .Therefore,
#f^((n))(0)=a_n n!# This reasoning can be extended to Taylor series around
#x_0# , of the form:#sum_(n=0)^(oo) c_n (xx_0)^n# With the relation
#f^((n))(x_0)=c_n n!# It's important to emphasize that the function
#n# th derivative of#f# (that is,#f^((n)) (x)# ) cannot be obtained directly from the Taylor/MacLaurin series (only it's value on the point around wich the series is constructed).
Questions
Power Series

Introduction to Power Series

Differentiating and Integrating Power Series

Constructing a Taylor Series

Constructing a Maclaurin Series

Lagrange Form of the Remainder Term in a Taylor Series

Determining the Radius and Interval of Convergence for a Power Series

Applications of Power Series

Power Series Representations of Functions

Power Series and Exact Values of Numerical Series

Power Series and Estimation of Integrals

Power Series and Limits

Product of Power Series

Binomial Series

Power Series Solutions of Differential Equations