3] Prove that the sequence (x,) = (-1)"+' does not converge to any real number.

Q: Show that the sequence {cn} = {(−1)n 1/ n! } converges, and find its limit

A: The nth term of the given sequence is .

Q: Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0.

A: To prove that the relation between the limits of the two given sequences

Q: Prove that the sequence (xn) = (-1)"+1 [3 .does not converge to any real number

A:

Q: (a) If the sequence ()neN CX is convergent, show that it is bounded. (b) If the sequence (n)neN CX…

A: Thank you for the question. Since you have posted multiple questions we will solve the first…

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

A: Every bounded sequence has convergent subsequence.

Q: II. Prove if X and Y are the sequence of real number and converges to x a real number then product…

A: Given that X and Y are the sequence of real number And X and Y converges to x a  real number. The…

Q: 3. Sn = + + ...+ Show that the {sn} sequence given in general terms converges %3D

A:

Q: Find all numbers x such that the series n=1 converges!

A:

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

A:

Q: Determine the limit of the sequence or show that the sequence diverges. If it converges, find its…

A: we have to check sequence is convergent or divergent.

Q: Show that the sequence (x,) = () convergence.

A: We have to use limit n tends to infinity.

Q: The sequence a = n is converges to 0 o converges to 8 попe a divergent sequence

A:

Q: 2) Show that the sequence defined recursively by 3 4 An aj = 5 аn+1 satisfies 2 < an < 5 and is…

A:

Q: Suppose that anx" converges for x = 5. Must it also converge for x = 4? What about x = -3?

A:

Q: 5* Determine whether the sequence defined by a* %3D 8* +1 converges or diverges. If it converges,…

A:

Q: 3. Prove that the sequence sin(n) (0, 1). converges to n n+1

A: We calculate both limit separately as n tends to infinity.

Q: Find x so that x - 1, x, and x + 2 are consecutive terms of ageometric sequence.

A: Given x-1, x, x+2 are the consecutive terms of a geometric sequence. To find x Let t1=x-1t2=xt3=x+2…

Q: Use differentiation to determine whether the sequence is strictly increasing or 2n +1 Sn=1 strictly…

A: We will check sequence is strictly increasing Or decreasing by using derivative.

Q: The series 1(-1)" converges to 0. True False

A: We have to find if the given series is converges.

Q: A sequence of functions that converges pointwise on [0, ∞0) but does not converge uniformly [0, 0).…

A:

Q: 6n Determine if the sequence an 4(-1) converges or diverges. If it converges, compute its limit. If…

A:

Q: Prove that the sequence (an) = (-1)"+' does not converge to any real number. %3D

A:

Q: 7. Determine whether the sequence, n 1+4n² 1+n² > converges or diverges. If it converges, find its…

A:

Q: Let (Xn)n and (Yn)n be two convergent sequences with limits x∗ and y∗. Prove that (a) xn + yn is a…

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Q: Analyse the convergence of the sequence {fn} where fu(x) = on R. 1+ na²

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Q: converges for all * > 0 because lim a - In n 0, for all * > 0- n=2 True False

A: Need to check given statement is true or false.

Q: 1. Use the definition of a convergent sequence to show {7 - converges to 7 where {7- is considered…

A: Convergent Sequence in Metric Space: Let be a metric space and let be a sequence in We say that…

Q: 1. Prove the sequence {a,} = (1 +-)" Converges to the limit e*, what about the sequence {b,} = (1…

A:

Q: prove that if X =CXn) is bounded increasing sequence of real ConVergent and Lim (xu) - Sup [An ineN]…

A:

Q: 3. Determine whether the sequence an = limit. x + 3 In x converges or diverges. If it converges,…

A:

Q: The sequence (tanh n) converges to2

A:

Q: 2. Use the definition to prove that if (xn) is a sequence of real numbers which converges to -3,…

A:

Q: A real sequence is defined as {xn} = n/(n+1) ,n : natural numbers. Using the formal definition of…

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Q: Prove that or 「+ X。- ズー 2. 71 number J2 Converged to

A:

Q: prove that the sequence (1/5^n) is convergent and approaches zero?

A: Series and sequences

Q: Show that the sequence ffn}, where fnox) = ne is uniformly convergent nx+1 on [a, b], aso but is…

A: Given: fnx=nxnx+1 To show that the given sequence is uniformly convergent on a,b, a&gt;0

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

A:

Q: The sequence (Xn) = (1, 3, 1, 3, 1, . . . , 1, 3, ...) is: Converge to 3. O Divergent. Converge to 0…

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Q: 7) Determine whether the sequence (1 - )} 3 4n converges or diverges. If it converges, find the…

A:

Q: (c) Prove that the sequence is Cauchy and conclude that it therefore converges.

A:

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

A:

Q: (3) Suppose {an}_1 is a sequence that converges to 0 and that an 2 0 for all n. Prove {Jan}1…

A:

Q: Show that n* 3-" converges for all exponents k. n=1

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

A:

Q: The sequence =n sin - converges Select one: True False

A:

Q: The sequence a = is a divergent sequence o converges to 8 none O converges to 0

A: Given: an=8nn For sequence, an, if limn→∞an+1an&gt;1, then sequence is divergent.

Q: 1. Show that the sequence ((nx - - 1/n)e n) converges uniformly on [0, 0). (15 points)

A:

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: The sequence an = n- -In+1vn +3 converges to the limit %3D A) -3 B) -1 C) 1 D) +2 E) -2

A:

Q: Find a sequence X, Such that X, does not converge (to any XE R) and Y, does converge to 0 where Y,…

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.