Which problem in mathematics is solved by Tylor series? Solution to which problem in mathematics is only given/solved by Tylor Series which is otherwise either impossible or extremely hard to solve?
What is the real advantage of Tylor's series ?
 A: This one is probably more on the the Physics side, but Perturbation Theory is definitely one of those problems. The idea is you know how to solve a system subject to an interaction $V_0$, and now want to know the solution if you change to $A$. To solve it, you expand
$$
A = A_0 + \epsilon + \epsilon A_1 + \frac{1}{2}\epsilon^2A_2 + \cdots
$$
and iteratively find the solutions at each order. Applications of this include scattering theory, Feynman Diagrams, ...
A: The Taylor Series is an expansion of the power series into an infinite sum of terms. 
For example: $e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$ 
The Taylor Series is used to get an approximate value of a function. The basic formula is f(x): $f(x) = f(a) +\frac{f'(a)}{1}(x-a) + \frac{f''(a)}{2!}(x-a)^{2} + \frac{f'''(a)}{3!}(x-a)^{3} + ... $ 
This formula is useful for 


*

*Approximating definite integrals of functions that have no definite integrals. 

*Understanding the growth of functions 

*Solving differential equations 


This website explains it in detail why the Taylor series works and how you get it.   http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx. 
