This can also be derived from a Power Series whose interval of convergence is (c-R, c+R) where R is a positive real number.

The Taylor series for f about c can be written as follows.

f(x) = f(c) + f^'(c)(x-c) +{ f^''(c)(x-c)²}/2! + {f^'''(c)(x-c)³}/3! + ... + { fⁿ(c)(x-c)ⁿ}/n! + ...

When c = 0, the Taylor Series is equivalent to Maclaurin Series.


Example:

Use Taylor Series to expand e^x around 1.

Solution:

f(x) = e^x and f(1) =e

f^'(x) = e^x and f^'(1)=e

f^''(x) = e^x and f^''(1) =e and so on.

Therefore, f(x) = e + e(x-1) + {e(x-1)²}/2! + {e(x-1)³}/3! + ...