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Numerical series
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1. A series is an infinite sum, written 
. It doesn’t have to start at k = 0.
2. The nth partial sum of the series is 
3. The series converges to a number L means: 
. In other words, for n large enough, 
is the same as L to any number of decimal places you like.
4. If converges to L, we sometimes write = L. In other words, the sum is as close to L as you want if you take enough terms.
5. Sometimes there is no number L with . In such a case we say that the series diverges. This could happen because the partial sums bounce around, or, in the case of a series like 
, because the numbers get bigger and bigger without bound.
6. Types of series:
a) is a positive term series means that 
for all k.
b) An alternating series has terms that switch signs. Every alternating series can be written as either 
or as 
where for all k.
c) A series is absolutely convergent if the positive term series 
is convergent.
If a series is absolutely convergent, it is convergent. The reverse might not be true. For example, if 
, then converges but is not absolute convergent, since = diverges. There is a name for this situation, as follows:
d) A series is conditionally convergent if converges but diverges.
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