Question

What is the difference between a sequence and a series?

Answer 1

See explanation...

Explanation:

A sequence is a list of values considered as individual terms.

A series is like a sequence, but instead of the terms being separate we are interested in their sum.

So an example sequence might be:

`1, 1/2, 1/4, 1/8, 1/16, 1/32,...`

with corresponding series:

`1+1/2+1/4+1/8+1/16+1/32+...`

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Footnote

More generally and formally, we could say that a sequence is a mapping from an index set `I` into a set of values `A`, where `I` is typically the set of positive integers `NN_1` or a finite subset of all positive integers up to a given value.

So a finite sequence of `4` numbers might be described as a mapping from `{1, 2, 3, 4}` to `RR` where:

`1 rarr 1`

`2 rarr 1/2`

`3 rarr 1/4`

`4 rarr 1/8`

We would describe this more briefly as the finite sequence:

`1, 1/2, 1/4, 1/8`

or say:

`a_1 = 1`, `a_2 = 1/2`, `a_3 = 1/4`, `a_4 = 1/8`.

An infinite sequence is (normally) a mapping from `NN_1` to a set `A`, which may be defined by a rule. For example:

`a_n = 2^(1-n)`

describes a mapping from `NN_1 -> RR` representing the sequence:

`1, 1/2, 1/4, 1/8, 1/16, 1/32,...`

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Advanced footnote

Why go into all of this formalism?

For one thing it allows us to generalise beyond countable sequences in order to prove results about larger infinite sets.

For a really scary example, see https://socratic.org/s/aBqGmvDC