Show that (1/n²+ n+1)!nEN is a Cauchy sequence.

Q: Find the generating function for the sequence 1,−2,4,−8,16,….

A: The general expansion of the binomial (1-x)-1 will be, (1-x)-1=1+x+x2+x3+….

Q: **) Define a sequence a1, a2, a3, ... as follows: a1 = 1, a2 = 3, and a, = an-1+ an-2 for al ategers…

A: The given sequence is 1,3,4,7,11,18,..... such that an=an-1+an-2 We have to prove P(n): an≤(74)n for…

Q: Find the generating function for the finite sequence 2, 2,2, 2, 2, 2.

A: Step 1: Consider the provided question, Given, the finite sequence 2, 2, 2, 2, 2, 2.

Q: Write the first five terms of the sequence defined recursively. a1 = 43,    ak + 1 = ak − 8 a1?…

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Q: Show X n² n² +1 is a Cauchy sequence using the definition.

A: We have to check weather the given sequence is cauchy sequence or not.

Q: Find an  for the sequence 1/1 -22/2x1 +33/3x2x1 - 44/4x3x2x1, +...

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Q: Consider the sequence {ax}, defined by а, 3D 1, ад — 1, and ak 3 а-1+ ak-2 for k > 3. Compute as.

A: We have given two terms of sequence {a_k}, a_1=1 and a_2=1 and then sequence {a_k} is defined by…

Q: Prove that the sequence {n³ + 1/n³} is a cauchy sequence.

A: The given sequence is an=n3+1n3. A sequence an is said to be Cauchy sequence if for each ε>0,…

Q: Prove {3n/(3n-1)} n=1 to infinity, is a decreasing sequence in R.

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Q: Let {n} be a sequence. Let Yn = if {n} is monotonic, 1. Show that n then so is {yn}.

A: What is Monotone Sequence: A sequence can either be constant or monotone. There are two different…

Q: Find the conditions necessary for this sequence to be Cauchy. Хп xn+1 = axn+b 1 + xn

A: For the sequence (Xn)(Xn​) defined by the recurrence relation to be Cauchy, the condition necessary…

Q: The sequence a,,a,a...... is defined as a₁ =n² +3n nez' i. Find the first four terms of the…

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Q: Calculate an for the sequence 2/(12+1) +1/(22+1) +2/(32+1) +1/(42+1) +.

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Q: Indicate the generating function of the sequence defined by ak = = 4ak-1 +5

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Q: Let ao, a1, a2, ... be the sequence defined recursively by an = 7(an-1)² if n ≥ 1 ao = 1, Use the…

A: Principle of mathematical induction states that a statement P(n) is true for all natural numbers n…

Q: Find the closed form of the generating function for the sequence 2,4, 8, 16, 32, ...

A: Generating Function: Let a0, a1, a2, a3, a4, .....be a sequence of real numbers denoted as an Then a…

Q: Find ai for the sequence with generating function, 1-x 1 x x

A: Simplify the generating function.

Q: Prove { (8n-7)/n} n=1 to infinity is an increasing sequence.

A: If we take the sequence as {Un}, and if we can able to show that for all n=1,2,3, ...  Un+1>Un…

Q: Show that the sequence delened by +5 mcreasing and ancg"for allkn an is thereas and anz9

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Q: If a sequence is unbounded, then it diverges. True or False?

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Q: Suppose b1, b2, b3, ... is a sequence defined as follows:  b1 = 4, b2 = 12,  bk = bk-2 + bk-1 for…

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Q: Suppose that b1, b2, b3, . . is a sequence defined as = 12, b = br–1 + b-2 for each integer k > 3.…

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Q: Consider the sequence recursively defined by bị = 3, b2 4, and bn+1 2b, + bn-1 for n > 2. Show that…

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Q: Suppose that f0, f1, f2, . . . is a sequence defined as follows:f0 = 5, f1 = 16,fk = 7fk−1 − 10fk−2,…

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Q: Find the generating function for the finite sequence 1, 4,16, 64, 256.

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Q: Show that the sequence {1, -1, 1, -1, 1, -1, ..., (-1)n+1, ...} diverges.

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Q: Suppose that f0, f1, f2, ...  is a sequence defined as follows:  f0 = 5, f1 = 16, fk = 7fk-1 -…

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Q: Consider the sequence {ak}1 defined by a1 = 1, and ak = ak-1+ 2(k – 1) for k > 2. Compute a5.

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Q: Find the first 10 terms of the sequence An 2n + 3. its 9th term is its 10th term is

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