Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 9



Series

We have seen in the last lecture that one can consider a number in the decimal numeral system, meaning an (infinite) sequence of digits between and , as an increasing sequence of rational numbers. For this, the -th digit after the separator, namely , means and has to be added to the approximation given by the digits before. The sequence of digits describes with the inverse powers of the difference between the approximating sequence, and the members in the approximating sequence are gained by summing up these differences. This viewpoint leads to the concept of a series.


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

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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.
  1. The series given by is also convergent and its sum is .
  2. 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

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holds.

Proof



Lemma

Let

denote a convergent series of real numbers. Then

Proof  

This follows directly from Lemma 9.4 .


Nikolaus of Oresme (1330-1382) proved that the harmonic series diverges.

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

So this series is about the "infinite sum“ of the unit fractions

This series diverges: For the numbers , we have

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Therefore,

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Hence, the sequence of the partial sums is unbounded, and so, due to Lemma 9.10 , not convergent.

The divergence of the harmonic series implies that one can construct with equal building bricks an arbitrary large overhang.

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.

Proof

This proof was not presented in the lecture.



Absolutely convergent series

Definition  

A series

of real numbers is called absolutely convergent, if the series

converges.

Lemma

Proof  

Let be given. We use the Cauchy-criterion Since the series converges absolutely, there exists some such that for all the estimate

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holds. Therefore,

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which means the convergence.



Example

A convergent series does not in general converge absolutely, the converse of Lemma 9.9 does not hold. Due to the Leibniz criterion the alternating harmonic series

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converges, and its sum is , a result we can not prove here. However, the corresponding absolute series is just the harmonic series, which diverges due to Example 9.6 .

The following statement is called the direct comparison test.


Lemma

Let be a convergent series of real numbers and a sequence of real numbers fulfilling for all . Then the series

is absolutely convergent.

Proof  

This follows directly from the Cauchy-criterion



Example

We want to determine whether the series

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converges. We use the direct comparison test and Example 9.2 , where we have shown the convergence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle {{}} \sum_{k <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 9]] __NOINDEX__ 1}^n \frac{ 1 }{ k(k+1) }} . For we have

Hence, converges and therefore also . This does not say much about the exact value of the sum. With much more advanced methods, one can show that this sum equals .



Geometric series and ratio test
This image illustrates the behavior of the geometric series for . Suppose that the length of the square is . Then we can put the geometric series threetimes inside this square. The area of the three series is .

The series is called geometric series for , so this is the sum

The convergence depends heavily on the modulus of .


Theorem

For all real numbers with , the geometric series converges absolutely, and the sum equals

Proof  

For every and every we have the relation

and hence for the partial sums the relation (for )

holds. For and this converges to because of Lemma 8.1 and exercise *****.


The following statement is called ratio test.


Theorem

Let

be a series of real numbers. Suppose there exists a real number with , and a with

for all (in particular for ). Then the series converges absolutely.

Proof  

The convergence does not change (though the sum) when we change finitely many members of the series. Therefore, we can assume . Moreover, we can assume that all are positive real numbers. Then

Hence, the convergence follows from the comparison test and the convergence of the geometric series.



Example

The Koch snowflakes are given by the sequence of plane geometric shapes , which are defined recursively in the following way: The starting object is an equilateral triangle. The object is obtained from by replacing in each edge of the third in the middle by the corresponding equilateral triangle showing outside.

Let denote the area and the length of the boundary of the -th Koch snowflake. We want to show that the sequence converges and that the sequence diverges to .

The number of edges of is , since in each division step, one edge is replaced by four edges. Their length is of the length of a previous edge. Let denote the base length of the starting equilateral triangle. Then consists of edges of length and the length of all edges of together is

Because of , this diverges to .

When we turn from to , there will be for every edge a new triangle whose side length is a third of the edge length. The area of an equilateral triangle with side length is . So in the step from to there are triangles added with area . The total area of is therefore

If we forget the and the factor , which does not change the convergence property, we get in the bracket a partial sum of the geometric series for , and this converges.


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