What is a partial sums?

Answer 1

Partial sums are used in calculus: for integration.

Suppose you wish to integrate an algebraic fraction of the form f(x)/g(x) where f and g are polynomial functions of x with integer coefficients. If the coefficients are rational but not integer then they can be converted to integer simply by using the LCM.

Also, suppose that the order of f(x) is less than that of g(x). Otherwise divide f(x) by g(x) to reach a position where the order of f is less.

Then if g(x) can be factorised as g(x) = p(x)*q(x) where p and q are polynomials in x and are of lower order than g.

Then f(x)/g(x) can be written as u(x)/p(x) + v(x)/q(x) where u and v are of lower order than p and q, respectively.

This is particularly important when g(x) is a quadratic and g(x) = p(x)*q(x) where p and q are binomials.

Then f(x)/g(x) = A/p(x) + B/q(x) where A and B are constants.

and then, if I represents the integral,

I(f/g) = I(A/p) + I(B/q) = A*I(1/p) + B*I(1/q) = A*ln|p| + B*ln|q| + C where C is the constant of integration.

= k*ln{|p|^A/|q|^B} where k = e^C

u(x)/p(x) + v(x)/q(x) are partial fractions for f(x)/g(x). Because partial fractions are polynomials of lower order than the original fractions, there is a greater chance that there is a simple analytical integral.

Answer 2

Partial sum is a sum of part of the infinite series. However, series is called a sum of all the terms in infinite series. Hence partial sum is a finite series.