B) If the partial sums s, of Ea, are bounded. Show that converges to n(n+1)

Q: 3. Suppose {fa) is an equicontinuous sequence of functions on a compact set K, and U.) converges…

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Q: Consider the sequence n²x(1 – nx) x € [0, 1/n] { - fn(x) := elsewhere . Does it converge uniformly…

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Q: Use the Monotone Convergence Theorem to show that the sequence n! +∞o (1 · 3 · 5 ··· (2n − 1)) n = 1…

A: l

Q: C. Working directly from the definition (and without using the Cauchy Completeness Theorem), show…

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Q: {xn} is a bounded sequence such that all convergent subsequences converge to the same number

A: Every bounded sequence has convergent subsequence.

Q: Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1

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Q: 7. a) Let x = c, with ||- Give an example of a Cauchy sequence in X that do not converge in X.…

A: Here, X=C00  is set of all sequence having only finitely many non zero terms with the norm xp where…

Q: Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1].

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Q: 4. Let f: DCR→ R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that {f(Xn)}…

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Q: Find the interval of convergence of 5 (x - 9)" In (5n) n=2

A: Solution of question is written in below steps.

Q: Q4/ Use Partial Sum Test to check the convergence of the following se 00 (-1)" n=0 Ans./ Diverged

A: Topic- convergence divergence

Q: Prove that {x_n} converges. Need to use info in a) and b)

A: We have the following information from part(a) and part(b) for any a,b∈ℝ…

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Q: 33. Show that 1 3 3 4 converges by computing the partial sums. Does it converge absolutely?

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Q: Prove that the sequence {n} defined by Xn = 1+ is a Cauchy sequence.. ¹ + 1 + 1 + 4 9 +

A: Introduction: A sequence is called a Cauchy sequence when its terms eventually become arbitrarily…

Q: Σ Show that n2 + 1 converges/diverges n

A: The given series is: ∑n=1∞nn2+1 Let  an=nn2+1,n≥1    =nn21+1n2   =1n1+1n2 And, bn=1n

Q: If the infinite sum 7 k=2 k2 converges, use the Remainder Estimate to find lower and upper estimates…

A: Given ∑K=2∞1K72

Q: ) Prove that if XnXnand YnYn are any two sequences of real numbers such that XnXn converges to xx…

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Q: 4. Show that the sequence {fn}, where fn(x) = xER 1+n°x*' is point-wise convergent but is not…

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Q: n En=0m! I)Use Cauchy criterion prove that the sequence does not converge uniformly

A: Given: ∑n=0∞xnn!

Q: 4. Let f : DCR → R be uniformly continuous and {In} be a Cauchy sequence in D. Then show that {f(1)}…

A: Image of a Cauchy sequence

Q: If 3 -1 series converges is the sequence of partial sums for the series 2 a,, then n=1 Select one:…

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Q: 1) Use the rigorous definition of convergence (in other words, an epsilon argument) to prove that…

A: We have to prove that sequence xn  is convergence using rigorous definition for convergence sequence

Q: { n! 1.3.5...(2n-1) +8 n = 1

A: This is the question of the sequence of real numbers.

Q: n! 1 · 3 · 5 ··· (2n − 1). +∞ n=1

A:

Q: 8. Show that the sequence f,(2) =. converges uniformaly in the region 1+ na l2l2 2, but does not…

A: Show that the sequence fnz=11+nz converge suniformly in the regionz≥2, but does not converges…

Q: compute the partial sum when k=1 goes until infinity of ak. ak = 1/((k^2)+3k+2)

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Q: n2 +1

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Q: Let (an) be a sequence of nonnegative real numbers. Prove that ∑an converges iff the sequence of…

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Q: 4. Let f: DCR → R be uniformly continuous and {r} be a Cauchy sequence in D. Then show that {S(rn)}…

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Q: Let (rn) be a sequence of real numbers. Show that the convergence of (sn) implies theconvergence of…

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Q: 27 n! n=0 converges to 0. 8.

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Q: Is it true that a sequence {an} of positive numbers must converge if it is bounded from above? Give…

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Q: 9 Use the monotone convergence theorem to determine the limit of the following sequence. 2an with a1…

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Q: (1) Show that n2 +1 n=1 converges conditionally. Show that all conditions of the test used are…

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Q: If Ea, converges absolutely, prove that Ea,² converges.

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Q: Let an n > 3. nvInn (b) Without computing any limits, use the Monotone Convergence Theorem to…

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Q: Exercise 9. Let {an}_1 be a bounded sequence and {b„}=1 be a sequence converges to 0. Prove that the…

A: Given that an is bounded. Then there exists M>0, such that an≤M for all n. Thus we get  anbn≤Mbn…

Q: 4. Show that the sequence { fn}, where nx fa(x) = xER 1+n°x2 is point-wise convergent but is not…

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Q: 4. Let f : DCR R be uniformly continuous and {r,} be a Cauchy sequence in D. Then show that {/(xn)}…

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Q: If 21ak converges then E,ak, converges for any ak subsequence of ak.

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Q: n! 1.3.5•••(2n-1) -} +∞ n=1

A: The question is solved by monotone convergence theorem. The sequence is decreasing and bounded below…

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Q: If the sequence (xn) convergence to x, then the sequence (VXn ) also convergence .x to إختر واحدا:…

A: If the sequence (xn) is convergent to x then the sequence sqrt(xn) is convergent to sqrt(x). Thus,…

Q: Use the Monotone Convergence Theorem to show that the following sequence 1.3.5(2n – 1)1 2 .4 . 6 ...…

A: Given: 1.3.5...(2n-1)2.4.6...(2n)n=1+∞

Q: 4. Show that the sequence (n) does not converge uniformly on [0, 2] by showing that the limit…

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