What is the difference between a sequence and a series?

1 Answer
Jan 18, 2017

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