Infinite Series

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Series

Note that this is not the same as a sequence because we are interested in the sum of the items. Also see summation notation.

Arithmetic series

An arithmetic series is the sum of an infinite number of terms that have a constant difference between successive terms

Example

Equation

The equation for finding the sum of an arithmetic series is as follows:

Where is the number of terms, is the start term, and is the nth term, or last term

Geometric series

Unlike Arithmetic series, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms

Example

It's a fact that . If , then .

Equation

The equation for finding the sum of a geometric series is as follows:

where if the first term, is the common ratio, is the number of terms, and is the sum of the first terms.

Harmonic Series

Definition

The harmonic series is formed by summing all positive unit fractions:

The sums of the terms approach infinity, so the harmonic series diverges (i.e does not add up to any particular number).