= Series =

A '''series''' is the sum of a finite or infinite '''sequence'''.

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== Description ==

A sequence can be finite or infinite. In the former case, these are often expressed with literal values like ''{1, 2, 3}''. In the latter case, a builder syntax is most often used; for example ''{1/n}''. If the trend is obvious, an infinite sequence could be written like ''{1, 2, 3, ...}''.



=== Arithmetic Sequences ===

An '''arithmetic sequence''' can be expressed as ''a,,n,, = a,,1,, + (n - 1)d'' where ''d'' is the step size and ''a,,0,,'' is the first (''n = 1'') term.

An infinite arithmetic series is always infinite.

However, the partial sum (i.e., cumulative up to ''n'' terms) can be calculated as:

{{attachment:arith.svg}}



=== Geometric Sequences ===

A '''geometric sequence''' can be expressed as ''a,,n,, = a,,1,,r^(n-1)^'' where ''r'' is the rate and ''a,,0,,'' is the first (''n = 1'') term.

An infinite geometric series only converges if ''|r| < 1''. (Note that if ''|r| = 1'' then the sequence is not actually geometric.) In the case that the series converges, it is calculated as:

{{attachment:geomconv.svg}}

A partial sum is calculated as:

{{attachment:geompart.svg}}

In some cases, a geometric sequence is instead formulated as ''a,,n,, = ar^n^''. Note that ''a'' is '''not''' the same as ''a,,1,,'', so the first (''n = 1'') term should be calculated.



=== Notable Series ===

Some notable series to be aware of:

{{attachment:i.svg}}

{{attachment:i2.svg}}



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CategoryRicottone