# Real series/Introduction/Section

## Definition

Let be a
sequence
of
real numbers.
The *series* is the sequence of the *partial sums*

If the sequence
converges,
then we say that the *series converges*. In this case, we write also

for its limit,

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 but later.

## Example

We want to compute the series

For this, we give a formula for the -th partial sum. We have

This sequence converges to , so that the series converges and its sum equals .

## Lemma

Let

denote convergent series of real numbers with sums and

respectively. Then the following statements hold.- The series given by is also convergent and its sum is .
- For also the series given by is convergent and its sum is .

### Proof

## Lemma

Let

be a
series
of
real numbers. Then the series is
convergent
if and only if the following *Cauchy-criterion* holds: For every
there exists some such that for all

the estimate

holds.

### Proof

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.

## Example

The *harmonic series* is the series

This series diverges: For the numbers , we have

Therefore,

Hence, the sequence of the partial sums is unbounded, and so, due to fact, not convergent.

The following statement is called *Leibniz criterion for alternating series*.

## Theorem

Let be an decreasing null sequence of nonnegative real numbers. Then the series converges.