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Sequence and Summation Notation

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A sequence is an ordered set of numbers that may have a finite or infinite number of terms. If the sequence is finite, the last term is shown, like .

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For example, the numbers from 1 to 10 are a finite sequence: . A positive even number can be represented by , where is a positive integer, giving the infinite sequence .

The character "…" (called an ellipsis) means "keep going as before."

To avoid using up many different letters, often the same letter is used with a whole number to its right and below (called a subscript), like this: . Such an integer is called an index.

More compactly, sequence notation is used: means . If the number of terms is infinite, the sequence ends with "…", like this: .

A series is the sum of a sequence, for example, .

Like a sequence, the number of terms in a series may be finite or infinite.

The notation for a series with finitely many terms is , which stands for .

For infinitely many terms, the notation is , which stands for .

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Contributed by: S. M. Blinder (July 2018)
Open content licensed under CC BY-NC-SA

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A finite series always has a sum: just add up the terms. Formulas for two basic finite series are

and

.

An infinite series may or may not have a finite sum.

The partial sums of an infinite series are the sequence , , , …. If the sequence of partial sums has a limit, that is called the sum of the series.

The partial sums may grow without bound, like for the series . Then the sum is infinite and the symbol is used as if it were a number, like this:

.

Even if the terms get smaller, the partial sums may still grow without bound as more terms are added. For example,

.

An infinite series may have a finite sum, but the terms must get small quickly enough. Any decimal is the sum of an infinite series: the powers of 10 in the denominators grow so quickly. For example,

.

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