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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