Examples
The Maclaurin series for any polynomial is the polynomial itself.
The Maclaurin series for (1 − x)−1 for |x| < 1 is the geometric series
so the Taylor series for x−1 at a = 1 is
By integrating the above Maclaurin series we find the Maclaurin series for log(1 − x), where log denotes the natural logarithm:
and the corresponding Taylor series for log(x) at a = 1 is
and more generally, the corresponding Taylor series for log(x) at some is:
The Taylor series for the exponential function ex at a = 0 is
The above expansion holds because the derivative of ex with respect to x is also ex and e0 equals 1. This leaves the terms (x − 0)n in the numerator and n! in the denominator for each term in the infinite sum.
Read more about this topic: Taylor Series
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