Problem :
Compute the Taylor series for f (x) = 1/(1 + x).
The first few derivatives of the function are
f'(x)  =  

f''(x)  =  

f^{(3)}(x)  =  

so
f (0) = 1,
f'(0) =  1,
f''(0) = 2,
f^{(3)}(0) =  6. The general
case is clearly that
f^{(n)}(0) = ( 1)^{n}n!. Hence the Taylor series
for
f (x) is
p_{∞}(x)  =  x^{n} 

 =  ( 1)^{n}x^{n} 

 =  1  x + x^{2}  x^{3} + ^{ ... } 

Problem :
What is the Taylor series of a polynomial p(x) = a_{n}x^{n} + a_{n1}x^{n1} + ^{ ... } + a_{0}?
It is easy to check that the Taylor series of a polynomial is the polynomial itself!
(All the coefficients of higher order terms are equal to
0.)