Let be a
The series is the sequence of the partial sums
If the sequence
then we say that the series converges. In this case, we write also
and this limit is called the sum
of the series.
All concepts for sequences carry over to series if we consider a series as the sequence of its partial sums
Like for sequences, it might happen that the sequence does not start with
respectively. Then the following statements hold.
- The series given by
is also convergent and its sum is .
also the series given by
is convergent and its sum is .
real numbers. Then the series is
if and only if the following Cauchy-criterion holds: For every
there exists some such that for all
real numbers. Then
This follows directly from
It is therefore a necessary condition for the convergence of a series that its members form a null sequence. This condition is not sufficient, as the harmonic series shows.
The following statement is called Leibniz criterion for alternating series.
Let be an decreasing
real numbers. Then the
This proof was not presented in the lecture.