Sequences - Folgen

real sequence - reelle Zahlenfolge

a sequence of real numbers is made of an infinite number of elements in a fixed order. The index n of the number a1 states what position in the sequence the number has.

$${a_1,a_2,a_3, …}$$

definition

If a function f is defined over the domain (Definitionsmenge) N (not including zero) or an infinite subset (Teilmenge) of N, then f is a sequence (Zahlenfolge). The function value f(n) is written as an and is called the nth term of the sequence. The function f is written as (an).

examples

1, 2, 3, 4, …

2, 4, 6, 8, …

rule - Bildungsgesetz

the nth term is often found in an algebraic expression in terms of n. For example the square numbers

1, 4, 9, 16, …

are expressed as $${a_n = n^2}$$

recursive definition - rekursive Darstellung

Wenn man den n-ten Term mit Hilfe des vorherigen (n-1)-ten Termes berechnen kann, spricht man von einer rekursiven Darstellung der Zahlenfolge.

Hier ist ein Beispiel von einem Ball das von einer bestimmten Höhe h fallen gelassen wird und h1 die Höhe ist nach dem ersten Aufprall auf dem Boden.

$${ h_n = h_{n-1} \cdot {0,7}}$$

explicit definition - explizite Darstellung

Wenn aber der n-te Term direkt berechnet werden kann, nennt man diese Form eine explizite Darstellung der Zahlenfolge.

$${ h_n = h \cdot {0,7}^n}$$

geometric sequence - geometrische Folge

The above example is a geometric sequence where successive terms form a common ratio $${r = \frac{a_{n+1}}{a_{n}}}$$

more examples

1, 2, 4, 8, 16, … $${\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, … }$$

arithmetic sequence - arithmetische Folge

In an arithmetic sequence, successive terms are separated by a common difference. $${d = a_{n+1} - a_{n}}$$

and the general term of an arithmetic sequence is

$${a_n = a_1 + d \cdot (n-1)}$$

example

-1, 2, 5, 8, …

here, the first term is a1 = -1 and the common difference d = 3

so the rule (Bildungsgesetz) is

$${a_n = -1 + 3 \cdot (n-1)}$$

which can be simplified to the explicit definition

$${a_n = 3n-4}$$

the recursive definition is

$${a_n = a_{n-1}+3}$$

where a1 = -1

limits - Grenzwerte

when we investigate a sequence as n approaches infinity, and it seems to approach a fixed value, this is called a limit.

more …

series - Reihen

A series is a new sequence resulting from summing up the terms of a sequence.

For example, if we consider a geometric sequence defined as: $${a_{n+1} = r \cdot a_{n}}$$

where r is the common ratio,

the sum of this sequence is $${s_n = a_1 + a_1r + a_1r^2 + a_1r^3 … + a_1r^n}$$ factorising an gives $${\text{(I)} \space \space s_n = a_1 \cdot (1 + r + r^2 + r^3 … + r^n)}$$ multiplying the expression (I) with the factor r gives $${\text{(II)} \space \space r \cdot s_n = a_1 \cdot ( r + r^2 + …+ r^n + r^{n+1})}$$

subtracting (II) from I yields $${ s_n - r \cdot s_n = a_1 \cdot (1 + (r- r) + (r^2-r^2) + …+ (r^n-r^n) - r^{n+1})}$$ leaving us with $${ (1 - r) \cdot s_n = a_1 \cdot (1 - r^{n+1})}$$ dividing by (1 - r) $${ s_n = a_1 \cdot \left(\frac{1 - r^{n+1}}{ 1 - r}\right)}$$

geometric series where 0 < r < 1 - geometrische Reihe

example - snowflake

$${s_n = 1 + \frac{1}{3} + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 + … + \left(\frac{1}{3}\right)^n }$$ $${ = 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + … + \left(\frac{1}{3}\right)^n }$$ what happens as n tends to infinity? $${\lim\limits_{n \to \infty} s_n = \lim\limits_{n \to \infty} a_1 \cdot \left(\frac{1 - r^{n+1}}{ 1 - r}\right)}$$ here a1 is 1 and $${r = \frac{1}{3}}$$ substituting into the equation gives $${\lim\limits_{n \to \infty} s_n = \lim\limits_{n \to \infty} 1 \cdot \left(\frac{1 - (\frac{1}{3})^{n+1}}{ 1 - \frac{1}{3}}\right) = \frac{1 - 0}{ 1 - \frac{1}{3}} = \frac{3}{2}}$$

harmonic series - harmonische Reihe

$${s_n = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + … + \frac{1}{n} }$$

What happens as n tends to infintiy? In summation notation below using the Greek letter sigma (S).

$${s_n = \sum_{n = 1}^{\infty} \frac{1}{n}}$$

Does it converge? Does the series diverge?

proof by induction - Beweis durch Induktion

not required but quite useful

Centre for Inovation of Mathematics Teaching

Part 1 on sequences and series

Part 2 on sequences and series

mathematik-wissen.de - Mathematik für die Schule über Zahlenfolgen

www.lernhelfer.de on limits


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