The most important examples of series.
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Concept |
Content |
Absolute Convergence |
A strong concept of convergence of series. |
Root Criterion |
An important criterion to prove absolute convergence by means of the behavior of the n-th roots of the underlying sequence's terms. |
Comparison Test |
If a series converges can be checked with different tests. |
Quotient Criterion |
An important criterion to prove absolute convergence by means of ratios of the underlying sequence's terms. |
Before we give some criteria for the convergence of series, we first
present the probably most important series and analyze their
convergence.
Example 1.
For , the
geometric series is convergent if and only if .
Proof: We can show that the -th partial sum is given by Hence, is convergent if
and only if . In this case
we have
The harmonic series is divergent
to .
Proof: If we construct some unbounded subsequence
, the
divergence of the harmonic series is proven (since it is monotonically
increasing). Indeed, we now show the unboundedness of the subsequence
: First,
observe that
Now we take a closer look to the number : By definition of
, we have The inequality in the above
formula holds true since every summand is replaced by the smallest
summand . The second
last equality sign then comes from the fact that the number is summed up -times. Now using this inequality
together with the above sum representation for , we obtain
As a consequence, the subsequence is
unbounded.
For , the
sequence is convergent.
Proof: The sequence of partial sums is strictly
monotonically increasing due to
Therefore, by theorems about convergence of sequences, the convergence
of is shown
if we find some bounded subsequence . Again we use
the representation for as in example b). We
can estimate so we have for and, due to
, it holds that . Using that , we obtain
Hence, the sequence is bounded.
This implies the desired result.
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