Lagrange Form of the Remainder Term in a Taylor Series
Key Questions

By taking the derivatives,
#f(x)=e^{4x}#
#f'(x)=4e^{4x}#
#f''(x)=4^2e^{4x}#
.
.
.
#f^{(n+1)}(x)=4^{n+1}e^{4x}# So,
#R_n(x;3)={f^{(n+1)}(z)}/{(n+1)!}(x3)^{n+1}={4^{n+1}e^{4z}}/{(n+1)!}(x3)^{n+1}# ,
where#z# is between#x# and#3# . 
Remainder Term of Taylor Series
#R_n(x;c)={f^{(n+1)}(z)}/{(n+1)!}(xc)^{n+1}# ,where
#z# is a number between#x# and#c# .
Let us find
#R_3(x;1)# for#f(x)=sin2x# .By taking derivatives,
#f'(x)=2cos2x#
#f''(x)=4sin2x#
#f'''(x)=8cos2x#
#f^{(4)}(x)=16sin2x# So, we have
#R_3(x;1)={16sin2z}/{4!}(x1)^4# ,where
#z# is a number between#x# and#1# .
I hope that this was helpful.

Taylor remainder term
#R_n(x;c)={f^{(n+1)}(z)}/{(n+1)!}(xc)^{n+1}# ,where
#z# is between#x# and#c# .
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