Give an example of a sequence (a) with (a0, but an divergent example of a sequence...
(c) (5 marks) Give an example of i. a sequence of real numbers that is strictly increasing and converges to zero; ii. a sequence of real numbers that is not monotonic and converges to 2 iii. a sequence of real numbers that is bounded and divergent. (d) (5 marks) Calculate the first four terms in the Laurent series representation of e*.
6. Give an example of a non-constant sequence that satisfies the given conditions or explain why such a sequence does not exist: (1) {an} is bounded above but not convergent. (2) {an} is neither decreasing nor increasing but still converges. (3) {an} is bounded but divergent. (4) {an} is unbounded but convergent. (5) {an} is increasing and converges to 2.
Series:
Is this example divergent or convergent.
Show using appropriate tests.
Σ()
Is the following series cos n convergent or divergent? Prove your result. 2 if Σ an with an > o is convergent, then is Σ a.. always convergent? Either prove it or give a counter example. 3 Is the following series convergent or divergent? if it is divergent, prove your result; if it is convergent, estimate the sum. 4 Is the following series 2n3 +2 nal convergent or divergent? Prove your result.
problem 3 and 4
Problem 3: Prove that if a → oo then a is divergent. Problem 4: Give an example of a sequence that is not bounded and does not diverge to oo.
2. Is the following sequence (cos(n))men properly divergent or divergent? Substantiate your result with a rigorous proof.
give an example of an arithmetic sequence that is found in the real world. find the common difference and write a recursive and iterative rule for the sequence. then give an example of a geometric that is found in the real world. find the common ratio and write ac recursive and iterative rule for the sequence. use a rule to find any term.
give an example of an arithmetic sequence that is found in the real world. find the common difference and write a recursive and iterative rule for the sequence. then give an example of a geometric that is found in the real world. find the common ratio and write ac recursive and iterative rule for the sequence. use a rule to find any term.
Describe in pseudocode an algorithm that takes as input a sequence of distinct numbers a0, a1, ..., an and returns 1 if the sequence is ordered increasingly, -1 if the sequence is in decreasing order, and 0 otherwise. What are the worse-case, best-case, and avergae-case of the algorithm? Analyze the running time in each of the three cases using asymptotic notation.
5. Give an example of a bounded sequence {sn)1 such that the set con- sisting of all its subsequential limits is precisely the closed interval [0, 1]. (Prove that your example has this property).