What is the Difference Between a Sequence and a Series?

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While the English words "sequence" and "series" have similar meanings, in mathematics they are completely different concepts. A sequence is a list of numbers placed in a defined order while a series is the sum of such a list of numbers. There are many kinds of sequences, including those based on infinite lists of numbers. Different sequences and the corresponding series have different properties and can give surprising results.

TL;DR (Too Long; Didn't Read)

Sequences are lists of numbers placed in a definite order according to given rules. The series corresponding to a sequence is the sum of the numbers in that sequence. Series can be arithmetic, meaning there is a fixed difference between the numbers of the series, or geometric, meaning there is a fixed factor. Infinite series have no final number but may still have a fixed sum under certain conditions.

Types of Sequences and Series

Common sequences are arithmetic or geometric. In an arithmetic sequence, each number or term of the sequence differs from the previous term by the same amount. For example, if an arithmetic sequence difference is 2, a corresponding arithmetic sequence might be 1, 3, 5.... If the difference is -3, a sequence might be 4, 1, -2.... The arithmetic sequence is defined by the starting number and the difference.

For geometric sequences, the terms differ by a factor. For example, a sequence with a factor of 2 might be 2, 4, 8... and a sequence with a factor of 0.75 might be 32, 24, 18.... The geometric sequence is defined by the starting number and the factor.

The series types depend on the sequence that is being added. An arithmetic series adds the terms of an arithmetic sequence, and a geometric series adds a geometric sequence.

Finite and Infinite Sequences and Series

Sequences and the corresponding series can be based on a fixed number of terms or an infinite number. A finite sequence has a starting number, a difference or factor, and a fixed total number of terms. For example, the first arithmetic sequence above with eight terms would be 1, 3, 5, 7, 9, 11, 13, 15. The first geometric sequence above with six terms would be 2, 4, 8, 16, 32, 64. The corresponding arithmetic series would have a value of 64 and the geometric series 126. Infinite sequences don't have a fixed number of terms, and their terms can grow to infinity, decrease to zero or approach a fixed value. The corresponding series can also have an infinite, zero or fixed result.

Convergent and Divergent Series

Infinite series are divergent if the sum approaches infinity as the number of terms increases. An infinite series is convergent if its sum approaches a non-infinite value such as zero or another fixed number. Series are convergent if the terms of the underlying sequence rapidly approach zero.

The series adding the terms of the infinite sequence 1, 2, 4... is divergent because the terms of the sequence keep growing, allowing the sum to reach an infinite value as the number of terms increases. The series 1, 0.5, 0.25... is convergent because the terms rapidly become very small.

While sequences are ordered lists of numbers and series are sums, both can be important tools in evaluating sets of numbers, and the properties of convergence or divergence may have real life implications. A divergent series often represents an unstable condition while a convergent series often means that a process or structure will be stable.

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