Series Expansions Math

A power series based on a function s derivatives at a single point.

Series expansions math. S n difference between successive terms. S arithmetic series formulas. A a n dn 1 1 1 1 2 i i i a a a 1 2 n n a a s n 2 11 n 2. Sin x x x3 3.

If a function f x has continuous derivatives up to n 1 th order inclusive then this function can be expanded in a power series about the point x a by the taylor formula. For 1 x 1 1 cn x k 1 1 2x 2 1 24 1 4k 2 x 4. F n a x a n n. Asymptotic series otherwise asymptotic expansions are infinite series whose partial sums become good approximations in the limit of some point of the domain.

F a f a x a f a x a 2 2. Q sum to infinity. Geometric series k 1 n k z k z 1 n 1 z n n z n 1 1 z 2 displaystyle sum k 1 n kz k z frac 1 n 1 z n nz n 1 1 z 2 k 1 n k 2 z k z 1 z n 1 2 z n 2 n 2 2 n 1 z n 1 n 2 z n 2 1 z 3 displaystyle sum k 1 n k 2 z k z frac 1 z n 1 2 z n 2n 2 2n 1 z n 1 n 2 z n 2 1 z 3. The kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.

Here are series expansions some maclaurin some laurent and some puiseux for a number of common functions. A later landmark in indian mathematics was the development of the series expansions for trigonometric functions sine cosine and arc tangent by mathematicians of the kerala school in the 15th century ce. A n number of terms in the series. In the 17th century james gregory also worked in this area and published several maclaurin series.

In general they do not converge but they are useful as sequences of approximations each of which provides a value close to the desired answer for a finite number of terms. An extension of the taylor series allowing negative exponent values. Cos x 1 x2 2. The resulting series often can be limited to a finite number of terms thus yielding an approximation of the function.

Try using 2 n fact n and n 0 to 20 in the sigma calculator and see what you get. A 1 nth term. A special case of a taylor series centred at zero. A series expansion is a method for calculating a function that cannot be expressed by just elementary operators addition subtraction multiplication and division.

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 f x. Arithmetic and geometric series definitions. There are several kinds of series expansions such as. N sum of the first n terms.