List of Maclaurin Series of Some Common Functions
- See also List of mathematical series
Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments x.
Exponential function:
Natural logarithm:
Finite geometric series:
Infinite geometric series:
Variants of the infinite geometric series:
Square root:
Binomial series (includes the square root for α = 1/2 and the infinite geometric series for α = −1):
with generalized binomial coefficients
Trigonometric functions:
Hyperbolic functions:
The numbers Bk appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. The Ek in the expansion of sec(x) are Euler numbers.
Read more about this topic: Taylor Series
Famous quotes containing the words list of, list, series, common and/or functions:
“Modern tourist guides have helped raised tourist expectations. And they have provided the natives—from Kaiser Wilhelm down to the villagers of Chichacestenango—with a detailed and itemized list of what is expected of them and when. These are the up-to- date scripts for actors on the tourists’ stage.”
—Daniel J. Boorstin (b. 1914)
“Lovers, forget your love,
And list to the love of these,
She a window flower,
And he a winter breeze.”
—Robert Frost (1874–1963)
“Autobiography is only to be trusted when it reveals something disgraceful. A man who gives a good account of himself is probably lying, since any life when viewed from the inside is simply a series of defeats.”
—George Orwell (1903–1950)
“This fond reiteration of the oldest expressions of truth by the latest posterity, content with slightly and religiously retouching the old material, is the most impressive proof of a common humanity.”
—Henry David Thoreau (1817–1862)
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)