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1 Convergent sequences and continuous functions
Englische Aufgaben zum Kapitel 1 Konvergente Folgen und stetige Funktionen
1.1
Decide whether the geometric sequence is strictly monotone increasing, decreasing or alternating.
Additionally, check the convergence of the sequence.
a. an = 36 ⋅
(31 )n
c. cn = 0.5 ⋅ (− 1.5 )n
b. bn = − 2 ⋅ 2.1n
d. dn = 2 ⋅
(32 )n
[strictly monotone … streng monoton; increasing … wachsend; decreasing … fallend; alternating … alternierend;
convergence … Konvergenz]
1.2 Compute the first 5 elements of the sequence. Have a guess about the monotonicity of the sequence
and prove your claim.
a. an =
n
n+ 4
b. bn =
+1
2 +n
n2 + 2
n2
n2 + 1
c. cn =
−3
[monotonicity … Monotonie; to prove … beweisen; claim … Behauptung]
1.3
Decide whether the sequence is convergent. Compute the limit c for any convergent sequence.
a. an =
3n 3 + 2n 2 + n
4 n 3 − 5n + 2
b. bn =
12 + n 2
n2 + 2
c. cn = 3 ⋅
−1
(101 )n
d. dn = −
4n 2
5n 2 + 1
+ 3n
[limit … Limes]
1.4 Compute the limit c and an index m such that | an − c | < 0.02 for all n ≥ m .
a. an = − 5n4n+ 1
b. bn =
n +5
2n + 6
c. c n =
3n
n2 + 1
for n > 0
1.5 Decide whether the geometric series with initial element a and quotient q is convergent and if so,
compute the limit.
a. a = 27; q =
5
2
b. a = -12; q =
3
3
c. a = 21 , q = 2
d. a = 21 , q =
π
6
[geometric series … geometrische Reihe]
1.6 Find a geometric sequence with initial element a and quotient q satisfying the specified properties.
∞
a. q = 61 ,
∑ a = 36
i
∞
b. a = 5 ,
i=0
∑ a = 25
i
c. q =
i=0
π
4
∞
,
∑ a =2 π
i
i=0
1.7 An abstract piece of art consists of wooden spheres placed next to one another. The radius of the first
sphere is 1m, the radius of each following sphere is 80% of the one before.
a. Compute the length of the line of 12 spheres.
b. Compute the volume of a line of infinitely many spheres.
c. Compute the radius of the initial sphere if the maximal length of the line of (infinitely many) spheres
is to be be 20m.
[piece of art … Kunstwerk; sphere … Kugel; infinitely … unendlich]
1.8 Compute the limit of the sequence bn and check the convergence of the images of the sequence with
respect to the given rational function f.
a. bn =
6n 2 − 3
2n 2 − n
, f ( x ) = 3x − 2
b. bn =
3n − 4
2 −n
, f( x ) =
4
x −2
+1
[image … Bild; with respect to … bezüglich; rational function … rationale Funktion]
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner
1 Convergent sequences and continuous functions
Englische Aufgaben zum Kapitel 1 Konvergente Folgen und stetige Funktionen
1.9 Which of these intervals contain a zero of f with f ( x ) = 4 x 3 − 35x 2 + 47 x + 14 ? Reason your decision.
A [− 9;−2
]
B [− 1; 0]
C [6; 8]
D [1; 1.5]
E [0; 3]
[interval … Intervall; zero … Nullstelle]
1.10 Decide, whether the shown functions are continuous. Name the points of discontinuity as appropriate.
a.
c.
b.
d.
[continuous … stetig; points of discontinuity … Unstetigkeitsstellen; as appropriate … gegebenenfallso … Nullstelle]
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner
1 Convergent sequences and continuous functions: solutions
Lösungen zu: Englische Aufgaben zum Kapitel 1 Konvergente Folgen und stetige Funktionen
1.1
a. an = 36 , 1, 4 , 43 ,  ; strictly monotone decreasing and convergent (0 < q < 1).
b. bn = − , − 4., − 8.8, − 18.5,  ; strictly monotone decreasing (a < 0) and divergent (q > 0).
c. c n = 0.5, − 0.75, 1.15, − 1.6875,  ; alternating and divergent (|q| > 0).
d. dn = , 3, 4.5, 6.75, 10.15,  ; strictly monotone increasing and divergent (q > 0).
1.2 a. an = 1, 1., 1.3 , 1.48  ,1.5,  ; guess: strictly monotone increasing; proof: We have to show that
an < an + 1 for all n * N. We get n +n 4 + 1 < nn++51 + 1 ⇒ n ⋅ (n + 5) < (n + 1) ⋅ (n + 4 ) ⇒ n2 + 5n < n2 + 5n + 4 . This
inequality holds for all n * N and thus, our claim is true, the sequence is strictly monotone increasing.
b. b = 1, 0.6 , 0.45, 0.3 ,0.59,  ; guess: strictly monotone decreasing; proof: We have to show that
n
an > an + 1 for all n * N. We get
2 +n
n2 + 2
>
3+n
n 2 + 2n + 3
2
⇒ ( 2 + n) ⋅ (n2 + 2n + 3) > (3 + n) ⋅ (n2 +2)
⇒ n3 + 4n2 + 7n + 6 > n3 + 3n2 + 2n + 6 ⇒ n > −5n . This inequality holds for all n * N and thus, our
claim is true, the sequence is strictly monotone decreasing.
c. cn = − 3, − .5, − ., − .1, − .059,  ; guess: strictly monotone increasing; proof: We have to
show that an < an + 1 for all n * N. We get
n2
n2 + 1
−3<
n 2 + 2n + 1
n 2 + 2n + 2
− 3 ⇒ n4 + 2n3 + 2n2 < n4 + 2n3 + 3n2 + 2n + 2 ⇒ 0 < n2 + 2n + 2 . This inequality holds
for all n * N and thus, our claim is true, the sequence is strictly monotone increasing.
[proof … Beweis; inequality … Ungleichung; claim … Behauptung]
1.3 a. convergent with limit c =
3
4
c. convergent with limit c = 0
b. convergent with limit c = 0
1.4 a. c = − 54 ; m = 8
d. divergent
b. c = 21 ; m = 48
c. c = 0; m = 150
1.5 a. divergent (q > 0)
c. divergent (q > 0)
b. convergent (since 0 < q < 1) with limit 0
1.6 a. a = 30, an = 30 ⋅
(61 )n
b. q =
4
5
d. convergent (since 0 < q < 1) with limit 0
, bn = 5 ⋅
(54 )n
c. a =
−π2 +4π
2
, cn =
− π2 +4π
2
⋅
(4π )n
1.7 a. length of the line of 12 spheres: ≈ 9.31 m
b. volume of a sphere: V = 43 r 3 π ; quotient for the sequence of volumina: q V = 0.83 ; volume of infinitely
many spheres: V ∞ = 4.2 m³
c. r = 4m
1.8 a. limit of bn : 3; f (bn ) is convergent with limit 7
b. limit of b n : −3 ; f (bn ) is convergent with limit
1
5
1.9 intervals containing a zero of f: B , C , E . f is a continuous function. According to the intermediate
value theorem, an interval [a; b] contains a zero of a continuous function if the images of a and b have
different signs. Since f is a polynomial function of degree 3, there are at most 3 zeros of f. Thus, the
other two intervals cannot contain any zeros of f.
[intermediate value theorem … Zwischenwertsatz; sign … Vorzeichen; degree … Grad]
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner
1 Convergent sequences and continuous functions: solutions
Lösungen zu: Englische Aufgaben zum Kapitel 1 Konvergente Folgen und stetige Funktionen
1.10 a. not continuous; points of discontinuity: 0; 3
b. continuous
c. not continuous; point of discontinuity: 2
d. continuous
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner