Mathematical Analysis

Sequences and Series

What is a series?

A series is closely related to the sum of numbers. It is actually used to help add numbers; therefore, in a sequence it can be the indicated sum of that sequence. In general, the idea is to start with a number, then do something to that number to get the next number, then do the same to that number to get the next number, and so on. For example, a finite series with six terms is 2 + 4 + 6 + 8 + 10 + 12, in which 2 is added to each number to get the next number. An example of an infinite series is one with the notation 1/2n, with n ≥ 1, or ½ + 1/4 + 1/8 + … (and so on).

To see a series written in notation, if set {xn} is a sequence of numbers being added, and set s1 = x1, then s2 = x1 + x2; s3 = x1 + x2 + x3; and so on. And for n ≥ 1, a new sequence is made, {sn}, called the sequence of partial sums, or sn = x1 + x2 + … + xn


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