Question

In this task, you will write a proof to analyze the limit of a sequence.

ASSUMPTIONS

Definition: A sequence {a_{n}} for n = 1 to ∞ converges
to a real number A if and only if for each ε > 0 there is a
positive integer *N* such that for all *n* ≥
*N*, |*a _{n}* –

Let *P* be 6. and Let *Q* be 24.

Define your sequence to be *a*_{n} = 4 +
1/(*Pn* + *Q*), where *n* is any positive
integer

A. Compute a positive integer *N* that is appropriate to
prove your sequence converges to a limit of 4. Show your work.

A.1. Using the formal definition provided in the Assumptions
section, prove, for ε > 0, that *a _{n}* converges
to a limit of 4. Justify your work.

Answer #1

In this task, you will write a proof to analyze the limit of a
sequence.
ASSUMPTIONS
Definition: A sequence {an} for n = 1 to ∞ converges
to a real number A if and only if for each ε > 0 there is a
positive integer N such that for all n ≥
N, |an – A| < ε .
Let P be 6. and Let Q be 24.
Define your sequence to be an = 4 +
1/(Pn +...

Assumptions:
The formal definition of the limit of a function is as follows:
Let ƒ : D →R with x0 being an
accumulation point of D. Then ƒ has a limit L at
x0 if for each ∈ > 0 there is a δ > 0
that if 0 < |x – x0| < δ and
x ∈ D, then |ƒ(x) – L| <
∈.
Let L = 4P + Q. when P = 6 and Q = 24
Define...

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lim (as n goes to infinity) 1/(n^2) = 0

Prove that a sequence (un such that n>=1)
absolutely converges if the limit as n approaches infinity of
n2un=L>0

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Analysing Algorithmic Efficiency (Marks: 3)
Analyze the following code fragment and provide an asymptotic
(Θ) bound on the running time as a function of n. You do not need
to give a formal proof, but you should justify your answer.
1: foo ← 0
2: for i ← 0 to 2n 2 do
3: foo ← foo × 4
4: for j ← 1396 to 2020 do
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6: foo ← foo ×...

You’re the grader. To each “Proof”, assign one of the following
grades:
• A (correct), if the claim and proof are correct, even if the
proof is not the simplest, or the proof you would have given.
• C (partially correct), if the claim is correct and the proof
is largely a correct claim, but contains one or two incorrect
statements or justications.
• F (failure), if the claim is incorrect, the main idea of the
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***Python
Hailstones, also known as the Collatz
sequence, are a mathematical curiosity. For any number in the
sequence, the next number in the sequence is determined by two
simple rules:
If the current number n is odd, the next number in the
sequence is equal to 3 * n + 1
If the current number n is even instead, the next
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n divided by 2)
We repeat this...

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