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) | = |
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 _{n-1} x ^{n-1} + ^{ ... } + 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 .)Problem : Find the Taylor series for the function g(x) = 1/ about x = 1 .
The first couple derivatives of the function are
g'(x) | = | x ^{-3/2} | |
g''(x) | = | x ^{-5/2} | |
g ^{(3)}(x) | = | x ^{-7/2} |
g ^{(n)}(1) = (1)(3)^{ ... }(2n - 1) = |
x ^{n} |