Given a sequence of numbers a_{1}, a_{2},…, a_{n},… (also denoted simply
{a_{n}}), we can form sums:
s_{n} = a_{1} + a_{2} + ^{ ... } + a_{n} 

obtained by summing together the first n numbers in the sequence. We call s_{n} the
nth partial sum of the sequence.
We would like to somehow define the sum of all the numbers in the sequence, if that is
something that makes any sense. We write this sum as
a_{n} = a_{1} + a_{2} + ^{ ... } 

and call it a series. In many cases, this sum clearly does not make sensefor
example, consider the case where we let each a_{n} = 1. As we add more and more of the
a_{n} together, the sum gets larger and larger, without bound. In other cases,
however, the sum of all the a_{n} seems to make sense. For example, let a_{n} = 1/2^{n}.
Then as we begin adding the a_{n} together, the sum looks like
As we add on more and more terms, the sum appears to get closer and closer to 1.
Let us make all of this a little more precise. Given a sequence {a_{n}}, the partial sums
s_{n} defined above as
s_{n} = a_{1} + a_{2} + ^{ ... } + a_{n} 

form another sequence, {s_{n}}. In our first example above, this sequence of partial
sums looks like
1, 1 + 1, 1 + 1 + 1, 1 + 1 + 1 + 1,… 

or
In our second example, the sequence of partial sums begins
If the terms of the sequence {s_{n}} gets closer and closer to a particular number as
n→∞, then we say that the series converges to L, or is convergent,
and write
a_{1} + a_{2} + ^{ ... } = a_{n} = s_{n} = L 

If the sequence of partial sums does not converge to any particular number, then we say
that the series diverges, or is divergent. Hence our first example above diverges and
our second example converges to 1; that is,
= 1 

As another example of a divergent series, consider the harmonic series:
To see that this sequence diverges simply note that a_{2}≥1/2, a_{3}, a_{4}≥1/4,
a_{5}, a_{6}, a_{7}, a_{8}≥1/8, etc. Thus,
s_{1}  ≥  1, 

s_{2}  ≥  1 + 1, 

s_{4}  ≥  1 + 1 +2, 

s_{8}  ≥  1 + 1 +2 +4 

and so on. We have s_{2n}≥1 + n/2, so the partial sums get arbitrarily large as
n→∞.
We conclude with two basic properties of convergent series. Suppose
a_{n} and b_{n} are two convergent series. Then
(a_{n} + b_{n}) also converges and
(a_{n} + b_{n}) = a_{n} + b_{n} 

Furthermore, if c is a constant, then ca_{n} converges and
ca_{n} = ca_{n} 
