Q. How do you show that a convergent sequence is bounded?
We are now going to look at an important theorem – one that states that if a sequence is convergent, then the sequence is also bounded. Theorem: If is a convergent sequence, that is $/lim_{n /to /infty} a_n = L$ for some , then is also bounded, that is for some , $/mid a_n /mid ≤ M$.
Q. Is every bounded sequence convergent example?
(a) Is every bounded sequence convergent? No. Here’s a counter-example: an=(−1)n⇝(−1,1,−1,1,−1,1,…)
Q. Can a sequence be bounded and convergent?
Theorem 2.4: Every convergent sequence is a bounded sequence, that is the set {xn : n ∈ N} is bounded. Remark : The condition given in the previous result is necessary but not sufficient. For example, the sequence ((−1)n) is a bounded sequence but it does not converge.
Q. What is a convergent sequence give two examples?
Mathwords: Convergent Sequence. A sequence with a limit that is a real number. For example, the sequence 2.1, 2.01, 2.001, 2.0001, . . . has limit 2, so the sequence converges to 2. On the other hand, the sequence 1, 2, 3, 4, 5, 6, . . . has a limit of infinity (∞).
Q. Are all monotonic sequences convergent?
A sequence (an) is monotonic increasing if an+1≥ an for all n ∈ N. The sequence is strictly monotonic increasing if we have > in the definition. Monotonic decreasing sequences are defined similarly. A bounded monotonic increasing sequence is convergent.
Q. Can a monotonic sequence diverge?
Monotonic Convergence Theorem: If a sequence is monotonic and bounded, if converges. Unboundedness Theorem: If a sequence is not bounded, it diverges.
Q. Does every increasing sequence diverge?
Every unbounded sequence is divergent.
Q. What does it mean for a sequence to be monotonic?
A sequence is said to be Monotone or Monotonic if it is either increasing or decreasing. A sequence is said to be Strictly Increasing if for all and Strictly Decreasing if for all . For example, consider the sequence. We note that , and so , and so this sequence is decreasing and hence monotone.
Q. How do you teach text structures?
Discuss with students that writers use text structures to organize information. Introduce the concept to them, and reinforce it every time students read and write. 2. Introduce and work on text structures in this order: description, sequence, problem and solution, cause and effect, and compare and contrast.
Q. What comes after the problem in a text?
What comes after the problem in a text? Words that help point out the problem and the solution in a text are called… quotes. It points out cause and effect in a text.
Q. What is the purpose of problem-solution?
Problem-Solution essays (or, as they may also be referred to, Proposing Solutions or Proposal essays) serve an important role. These essays inform readers about problems and suggest actions that could be taken to remedy these problems.
Q. What is the text structure of problem and solution?
Problem and Solution is a pattern of organization where information in a passage is expressed as a dilemma or concerning issue (a problem) and something that was, can be, or should be done to remedy this issue (solution or attempted solution).
Q. What is the cause effect text structure?
Cause and effect is a common way to organize information in a text. Paragraphs structured as cause and effect explain reasons why something happened or the effects of something. The cause and effect text structure is used so commonly that you have probably written a paragraph using it and not noticed.
