John Wiley and Sons. In the 17th century, James Gregory also worked in this area and published several Maclaurin series. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. If the Taylor series is centered at zero, then that series is also called a Maclaurin series , named after the Scottish mathematician Colin Maclaurin , who made extensive use of this special case of Taylor series in the 18th century. Applying the multi-index notation the Taylor series for several variables becomes.
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Series Expansion. A series expansion is a representation of a particular function as a sum of powers in one of its variables, or by a sum of powers of another (usually elementary) function. Here are series expansions (some Maclaurin, some Laurent, and some Puiseux) for a number of common functions. Cosine Taylor Series at 0 Derivation of the Maclaurin series expansion for cosine. This video can be found on the Kahn Academy website, and carries a Creative Commons copyright (CC BY-NC-SA ). Sine and Cosine: Expansions. Series: sin(x) = (-1) k x 2k+1 / (2k+1)! = x - (1/3!)x 3 + (1/5!)x 5 - (1/7!)x 7 (This can be derived from Taylor's Theorem.). cos(x.