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Finite and Infinite Mathematical Series

18 Apr Posted by in Algebra, Pre-Calculus | 179 comments

Overview:  What Is A Mathematical Series?

A mathematical series is the sum of the elements of a mathematical sequence.  If the series is finite, the sum will be a finite number.  For example, in the finite sequence {1, 2, 3, 4, 5}, the series is the sum of all the terms: 1 + 2 +3 +4 + 5 = 15.

Partial Sums and Sigma

Partial sums can be used, if the series is very long.  The final sum in the above example is easy to find, but it can also be approached by finding partial sums, such as the third partial sum in the series, S3, which is the sum of all the values of the first three numbers in the sequence, 1 +2 +3 or 6.  In formula terms, directions to find the sum are given by a Greek letter, sigma or ∑.

Infinite Mathematical Series

What is the mathematical sequence is infinite?  Finding the sum isn’t as easy as in a finite series, and the directions to find partial sums are very important.   For example, in the  series {2 + 4 +6 +… + 2n …} , the sum of the first term , S1=2, the second partial sum, S2, is 2 +4, or 6, and the third partial sum, S3, is 2 +4 +6 = 12.  The series will go on infinitely, because for every 2n, the nth term can always be one number larger.

Are All Mathematical Series Infinite?

The sum of many mathematical sequences is an infinitely large number, because n is infinitely large.  However, some mathematical sequences are infinite, but n becomes infinitely small, and the sum approaches a limit,  For example, suppose the mathematical sequence starts with a square with area 1, and then divide the square into halves.  The first rectangle will have an area 1/2 the size of the original square, the second rectangle will have an area 1/4 the size, the third will have an area 1/8, the fourth will have the area 1/16, and so on.  The pattern can be expressed as (1/2)1, (1/2)2, (1/2)3…(1/2)n…  If it is written as a series , then 1/2 +1/4 +1/8 +1/16 +… = 1, because the original square has area 1 and it was just divided into n rectangles.  The infinite series has a finite limit.

Real-World Applications Of Mathematical Series

Mathematical series have a number of useful applications.  For example, determining how much money will be earned with a recurring investment is an application of mathematical series used in banking.  There are a number of mathematical series that are used in radio and electronics, as well as in physics and computer science.

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