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series (Definition)

Given a sequence of numbers (real or complex) $ \{a_n\}$ we define a sequence of partial sums $ \{S_N\}$, where $ S_N=\sum_{n=1}^N a_n$. This sequence is called the series with terms $ a_n$. We define the sum of the series $ \sum_{n=1}^\infty a_n$ to be the limit of these partial sums. More precisely

$\displaystyle \sum_{n=1}^\infty a_n = \lim_{N\to\infty} S_n = \lim_{N\to\infty} \sum_{n=1}^N a_n. $
In a context where this distinction does not matter much (this is usually the case) one identifies a series with its sum, if the latter exists.

Traditionally, as above, series are infinite sums of real numbers. However, the formal constraints on the terms $ \{a_n\}$ are much less strict. We need only be able to add the terms and take the limit of partial sums. So in full generality the terms could be complex numbers or even elements of certain rings, fields, and vector spaces.



"series" is owned by mathwizard. [ full author list (2) | owner history (1) ]
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See Also: convergent series, harmonic number, complete ultrametric field, summation, prime harmonic series


Attachments:
sum of series (Definition) by pahio
remainder term (Definition) by PrimeFan
if $\sum_{k=1}^\infty a_k$ converges then $a_k\to 0$ (Theorem) by matte
arithmetic-geometric series (Derivation) by perucho
topic entry on series of complex terms (Topic) by pahio
real part series and imaginary part series (Theorem) by pahio
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Cross-references: vector spaces, fields, rings, even elements, complex numbers, strict, terms, infinite, partial sums, limit, sums, complex, real, sequence
There are 241 references to this entry.

This is version 4 of series, born on 2002-05-31, modified 2005-02-23.
Object id is 2973, canonical name is Series.
Accessed 16129 times total.

Classification:
AMS MSC: 40-00 (Sequences, series, summability :: General reference works )
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