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
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)