SERIESSUM(x, n, m, coefficients)

The SERIESSUM function calculates the sum of a power series:

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where coefficients is the list denoted by a1, a2, a3 …. aj.

Power series expansions are useful in approximating many functions. The function arguments are:

x

is a variable that contains the input value to the power series

n

is the initial power to which x is to be raised

m

is the step by which n is to be increased for each term of the series

coefficients

is a set of coefficients by which the successive powers of x are to be multiplied. The number of coefficients determines the number of terms in the power series.

For example, a power series can be used to approximate the function SIN(x). The power series expansion for SIN(x) is:

x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ….

The sequence of coefficients 1, -1/3!, 1/5!, -1/7! ….. can be written as

(-1)^0, ((-1)^1)/(m + n)!, ((-1)^2)/(2m + n)!, ((-1)^3)/(3m + n)! ….. where n = 1 and m =2.

Thus to calculate sin(45°) or, in radians, sin(p/4), set n = 1, m = 2, x = p/4, and put the coefficients that are to be used in a range. In a spreadsheet, if B1..B5 contain the first 5 coefficients (i.e. 1, -1/FACT(3), 1/FACT(5), -1/FACT(7), 1/FACT(9)), then the following formula can be used to approximate the sine function:

SERIESSUM(PI()/4, 1, 2, B1..B5)

which returns 0.707107.

The accuracy of the approximation depends, of course, on how many coefficients are used. The greatest accuracy possible is attained when the number of coefficients used is infinite! Note that computational modeling of mathematical functions such as sine and cosine is ultimately achieved by using power series expansions.

See also:

Other mathematical functions