MS-A0111 - Differential and Integral Calculus 1, Lecture, 5.9.2022-19.10.2022
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Differential and Integral Calculus
2. Series
Table of Content
Convergence
Convergence
If the sequence of partial sums has a limit
, then
the series of the sequence
converges
and its sum is
. This is denoted by
Divergence of a series
A series that does ot converge is divergent. This can happen in three different ways:
- the partial sums tend to infinity
- the partial sums tend to minus infinity
- the sequence of partial sums oscillates so that there is no limit.
In the case of a divergent series the symbol
does not really mean anything (it isn't a number). We can then interpret it as the sequence of partial sums, which is always well-defined.
Basic results
Geometric series
A geometric series
converges if
(or
), and then its sum is
. If
, then
the series diverges.
Rules of summation
Properties of convergent series:
Note: Compared to limits, there is no similar product-rule for series, because
even for sums of two elements we have
The correct generalization is the Cauchy product of two series, where also
the cross terms are taken into account.
Note: The property
cannot be used to justify the convergence of a series; cf. the following examples.
This is one of the most common elementary mistakes many people do when
studying series!
Example
Explore the convergence of the series
Solution. The limit of the general term of the series is
As this is different from zero, the series diverges.
This is a classical result first proven in the 14th century by Nicole Oresme after which a number of proofs using different approaches have been published. Here we present two different approaches for comparison.
i) An elementary proof by contradiction. Suppose, for the sake of contradiction, that the harmonic series converges i.e. there exists
such that
. In this case
Now, by direct comparison we get
hence following from the Properties of summation it follows that
But this implies that
, a contradiction. Therefore, the initial assumption that the harmonic series converges must be false and thus the series diverges.
ii) Proof using integral: Below a histogram with heights lies the graph of
the function
, so comparing areas we have
as
.
Positive series
Summing a series is often difficult or even impossible in closed form, sometimes only a numerical approximation can be calculated. The first goal then is to find out whether a series is convergent or divergent.
A series is positive,
if
for all
.
Convergence of positive series is quite straightforward:
Theorem 2.
A positive series converges if and only if the sequence of partial sums is bounded from above.
Why? Because the partial sums form an increasing sequence.
Example
Show that the partial sums of a superharmonic series
satisfy
for all
, so the series converges.
Solution. This is based on the formula
for
, as it implies that
for all
.
This can also be proven with integrals.
Leonhard Euler found out in 1735 that the sum is actually . His
proof was based on comparison of the series and product expansion of
the sine function.
Absolute convergence
Theorem 3.
An absolutely convergent series converges (in the usual sense) and
This is a special case of the Comparison principle, see later.
Suppose that converges. We study separately the positive and negative
parts of
:
Let
Since
, the positive series
and
converge by Theorem 2.
Also,
, so
converges as a difference of two convergent series.
Example
Study the convergence of the alternating (= the signs alternate) series
Solution. Since
and the superharmonic series
converges, then the original series is absolutely convergent. Therefore
it also converges in the usual sense.
Alternating harmonic series
The usual convergence and absolute convergence are, however, different concepts:
Example
The alternating harmonic series
converges, but not absolutely.
(Idea) Draw a graph of the partial sums to get the idea that
even and odd index partial sums
and
are monotone and
converge to the same limit.
The sum of this series is , which can be derived by integrating the
formula of a geometric series.
points are joined by line segments for visualization purposes
Convergence tests
Comparison test
The preceeding results generalize to the following:
Proof for Majorant. Since and
then
is convergent as a difference of two convergent positive series.
Here we use the elementary convergence property (Theorem 2.) for positive series;
this is not a circular reasoning!
Proof for Minorant. It follows from the assumptions that the partial sums of
tend to infinity, and the series is divergent.
Example
Solution. Since
for all
, the first series is convergent by the majorant principle.
On the other hand,
for all
, so the second series has a divergent harmonic series as a minorant.
The latter series is thus divergent.
Ratio test
In practice, one of the best ways to study convergence/divergence of a series is the so-called ratio test, where the terms of the sequence are compared to a suitable geometric series:
Limit form of ratio test
(Idea) For a geometric series the ratio of two consecutive terms is
exactly . According to the ratio test, the convergence of some other
series can also be investigated in a similar way, when the exact ratio
is replaced by the above limit.
In the formal definition of a limit . Thus starting from some index
we have
and the claim follows from Theorem 4.
In the case the general term of the series does not go to zero,
so the series diverges.
The last case does not give any information.
This case occurs for the harmonic series (, divergent!) and superharmonic
(
, convergent!) series. In these cases the convergence or divergence
must be settled in some other way, as we did before.