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 n th 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 sense--for 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
1, 2, 3, 4,… |
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} = c a _{n} |