Expert: Paul Klarreich - 7/17/2006

Question
what are the meanning of sequences and series

Answer
Hi, Dyunh,

Your Question:  what are the meaning of sequences and series
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Oh, is that all you have to know?  This shouldn't take more than a few months, but to get you started:

A SEQUENCE is a list of numbers, usually written in subscript notation, like:

a1, a2, a3, ..., an,...

A SERIES is derived from a sequence.  It is a new sequence that represents the partial sums of the original.

So, for example, if a sequence has these numbers:

a1 = 1
a2 = 3
a3 = 5
a4 = 7,
etc.

then the associated series is:

s1 = a1 = 1
s2 = a1 + a2 = 4
s3 = a1 + a2 + a3 = 9
s4 = a1 + a2 + a3 + a4 = 16

etc.

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and the basic thing we do with these things is to see if they converge.  That means:

Given a sequence or a series (they are not so different, after all)  a1, a2, a3, ..., an,...  we want to know if

    lim    [an]
n->infinity

or
    lim    [sn]
n->infinity

exists.

In the case of the above sequences, the answer is NO, they don't converge, they diverge.