taylor polynomial

The Taylor series of a real or complex function f(x) that is infinitely differentiable in a neighborhood of a real or complex number a, is the power series
圖片參考：http://upload.wikimedia.org/math/6/c/8/6c81bd7e1ffe0d67ce7a348f7fec25ea.png
which in a more compact form can be written as

圖片參考：http://upload.wikimedia.org/math/0/1/4/014b151d93eaf2312004358e25576ebf.png
where n! is the factorial of n and f (n)(a) denotes the nth derivative of f evaluated at the point a; the zeroth derivative of f is defined to be f itself and (x − a)0 and 0! are both defined to be 1.
Often f(x) is equal to its Taylor series evaluated at x for all x sufficiently close to a. This is the main reason why Taylor series are important.
In the particular case where a = 0, the series is also called a Maclaurin series.
f(y)=e^y
df(y)/dy = e^y
d^nf(y)/dy = e^y
for a = 0,
f(y) = f(0) + 1/1! df(0)/dy (y-0) + ... + 1/9! d^9f(0)/dy (y-0)^9
= e^0 ( 1 + 1/1! y + 1/2! y^2 + ... + 1/9! y^9 )
= ( 1 + 1/1! y + 1/2! y^2 + ... + 1/9! y^9 )

For y = 1, f(1) = e^1 = e
f(y) = ( 1 + 1/1! y + 1/2! y^2 + ... + 1/9! y^9 )
f(1) = ( 1 + 1/1! 1 + 1/2! 1^2 + ... + 1/9! 1^9 )
= 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + 1/8! + 1/9!

thx