Karlsruhe Institute of Technology
Institute for Algebra and Geometry
Prof. M. Axenovich Ph.D.
Dipl.-Math. P. Forster
21
22
23
24
25
P
Karlsruhe, November 15, 2012
Student No.:
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Worksheet No.5
Advanced Mathematics I
Exercise 21: Determine all accumulation points of the sequences (an )n and (bn )n with n ∈ N.
1
5n + 7 n
n
(a) an = + 2(−1)
i
(b) bn =
n
n
Exercise 22: Consider the following sequences given by
(−2)−n + 1
2n
−1+
, n ∈ N,
1 + 2n
1 + 2n
1 + 2n
, n ∈ N.
(d) dn =
1 + 2n + (−2)n
1 + 6n + 2n2
, n ∈ N,
(n + 3)n
6 + n2
(b) bn = 6 −
, n ∈ N,
n
(a)
(c)
an =
cn =
Examine these sequences for boundedness, monotonicity and convergence. (Limits or accumulation points need
not be specified). If appropriate, analyze suitable subsequences of each sequence whether they are bounded,
monotone, or convergent.
Exercise 23: For a ∈ R, a > 0, let the sequence (xn )n be defined recursively by
xn+1 = 2xn − ax2n , n ∈ N ∪ {0},
for fixed 0 < x0 < a1 .
(a) Show that xn+1 ≤
1
a
for every n ∈ N ∪ {0} (hint: completion of the square).
(b) Show by induction: xn ≥ 0 for every n ∈ N ∪ {0} (hint: use part (a)).
(c) Why does (xn )n converge? Determine lim xn .
n→∞
Exercise 24: Let the sequence (ak )k , k ∈ N, be defined recursively by
a1 = b ,
ak+1 =
|ak |
, k > 1,
2ak − 1
for two initial values b = − 41 , as well as b = 14 .
(a) Which are the possible limits under the assumption of convergence?
(b) For which initial value
1
4
or − 14 is the sequence monotonic?
(c) For which initial value is the sequence bounded?
(d) Does the sequence converge for either of the two initial values? If so, determine the limit.
Justify your answers to all these questions.
Exercise 25: A student memorizes three pages of AM1 lecture notes a day. Overnight he forgets 4% of his
total acquired knowledge. Assume that the lecture notes has infinite number of pages and that the student has
no AM1 knowledge at his first semester day.
(a) Set up the sequence (in a recursive form) for the amount of knowledge (wn )n , n ∈ N, (measured in pages)
of the student after n days and n nights.
(b) Prove that (wn )n is monotonically increasing.
(c) Prove that (wn )n is bounded above by 75 pages.
(d) What would the limit of (wn )n be?
Due date: Please hand in your homework on Monday, November 26, 12:00, into the AM1-box in the student
office in the International Department.
Tutorial 5
Advanced Mathematics I
Exercise T13: Which of the following assertions is true?
(a) A sequence converges, if it is monotonic and bounded.
(b) If a sequence converges, it is monotonic and bounded.
(c) If a sequence is not bounded, it can’t be convergent.
(d) If a sequence is not monotonic, it can’t be convergent.
(e) If a sequence has exactly one accumulation point, it converges.
(f) If a sequence converges, it has exactly one accumulation point.
Exercise T14: Analyze the following sequences whether they are bounded, monotone, or convergent (in case
of convergence you need not specify the limit):
(a) an =
1 + n + n2
, n ∈ N,
n(n + 1)
(b)
bn =
1 + n + n2
, n ∈ N,
n+1
(c)
1
, n ∈ N,
1 + (−2)n
(d) dn =
1 + (−2)n
, n ∈ N.
1 + 2n
cn =
Exercise T15: Let the sequence (ak )k , k ∈ N, be defined by
a1 = 1,
ak+1 =
a
pk
, k > 1.
1 + 1 + a2k
(a) Show that this sequence converges and compute its limit.
(b) Verify that the sequence (bk )k with bk = 2k ak , k ∈ N, converges as well.
Exercise T16: Let the sequence (an )n be recursively defined by the formula
1
an+1 = a2n + ,
4
n ∈ N ∪ {0}.
(a) Show that the sequence converges for every initial value 0 ≤ a0 ≤
(b) Show that the sequence diverges for each a0 >
1
2
1
2
and compute its limit.
(Hint: Show that an ≥ a0 + nd where d = (a0 − 1/2)2 ).
(c) Analyze the sequence (an )n whether it converges for initial value a0 < 0.
For detailed information regarding this course please check the page
http://www.math.kit.edu/iag6/lehre/am12012w/en
Tutorial date: Wednesday, November 21, 2012, 14:00-15:30 pm.