If (am) is increasing and en (an) is bounded by M converges to a, {|a1\, |a|}. max

Q: |an also converges because 0 If an converges then < lan|, for n = 1, 2, ... n=1 n=1

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Q: Show that the sequence {f}, where f₁²x) = tan `nx, x70 is uniformly ノ on any interval [a,b], aso but…

A: Given function as fnx=tan-1nx, x≥0

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

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

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Q: (b) Use the Direct Comparison or Limit Comparison Tests to test the following integral converges or…

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Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx Use the uniform norm to show that the…

A: Given: fn(x)=sinnx1+nx To show: fn(x) is uniformly converge on [1, ∞).

Q: 18 For which numbers a does x,+, = a{x,– x) converge to x* = (a – 1)/a?

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Q: Suppose {fn}=1 and {g„}=1 converges uniformly on E where {fn}=1 and {gn}n=1 are sequences of bounded…

A: Let fnn=1∞ is converges uniformly to f and gnn=1∞ is converges uniformly to g on E and fnn=1∞ and…

Q: Decide whether the following proposition is true or false. Provide a justification for your answer:…

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Q: The sequence an = n – vn+ lvn+3 converges to the limit

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Q: if Xn is bounded, is Xn  cauchy ?  is Xn convergent?

A: Solution

Q: The sequence an = n – Vn +1yn+3 converges to the limit

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Q: ounded funclions on A to h,t converges uniformly

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Q: Find sequences (xn) and (yn) so that xn - Yn 0 and yet does not converge to 1 Yn

A: By Bartleby policy I have to solve only first one as these are all unrelated problems.These are all…

Q: Suppose that dz converges. What can we conclude about the constant p? Op>1 Op Op< 3

A: Explanation of the answer is as follows

Q: an Show that if lim =0 and E-i b, converges then En=1 converges too. an =1 n+ 00

A: Given that  limn→∞anbn=0  and  ∑n=1∞bn converges.

Q: What are all of the values of x ,such that - (-1) n+A converges? n=1 .1 X26 .2 06

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

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Q: 4. Does cosnx uniformly converge in [a,b] c (0, 7)? n =1 n

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Q: For sequence of functions {nxe-nx} for x ∈ (0 + 1), what is the uniform norm of fn (x) - f(x) on (0…

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Q: If 2 a, converges, then la,| diverges. n=1 n=1 True False

A: Given statement is FALSE. I HAVE given a counter -example.

Q: Show that the sequence {fn} where fu(r) = e-n* is uniformly convergent on [a, b), a > 0.

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Q: Show that if each fn is bounded on A and (fn) converges uniformly to f on A, then f is bounded on A.

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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: #2. Give a counter exa mple that (Sn) is bounded but not convergent.

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Q: a) Let (h,). (t) be sequences of hounded functions on A that converges uniformly on A to h,t…

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Q: Two sequences of functions fn and gn that converge uniformly but fn In does not converge uniformly.

A: See the attachment

Q: If the interval of convergence for f(x) = E anx" is [-R, R], then an x" converges uniformly on [–R,…

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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: What are all of the values of x, such that converges?

A: We have to use GP sum Common Ratio.

Q: . Use the inegral test to show that =1 converges. 1+n2

A: To solve this question first we check derivative of function is negative for n&gt;0 and then we show…

Q: 3- a- By definition show that (x,) = (+3) converges to 3. b- Does (+3) bounded, Cauchy? Show by part…

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Q: By examining the limit as n→0, the sequence {2"}D=1 O Diverges O Converges to 0 1 Converges to

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Q: 2. Show that following sequence of functions are convergence pointwise to its limit function. a)…

A: See the attachment

Q: Let f : A → R and (xn),(Yn) are two sequences converges to c such that Xn, Yn # c, Vn E N. If…

A: We understand this statment by the help of an counterexample .

Q: The integral S x3 converges to Select one: O True False

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Q: Let fn(x) x +n b. Use the Lemma for non-uniform convergence to show that the sequence does not…

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Q: îf хи 1+nn } Convergent uniformy in [0,00) ?

A: Introduction: As a sequence of real numbers, a sequence of functions is also convergent or…

Q: Given two sequences (fn)neN, (9n) nEN C C[0, 1] of continuous functions on the closed unit interval…

A: Given two sequences fn, gn contained in the interval 0,1 of continuous function on 0,1 by…

Q: If \( \{a_n\}\)  is decreasing and bounded below then it is convergent.  true or false?

A: Monotone Convergence Theorem : Every bounded monotone sequence is convergent.

Q: Is the following statement correct: If (fn) is a sequence of differentiable functions on [a, b] such…

A: Given statement is

Q: Find the interval of convergence of this power serie (x+1)t

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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: Find, with justification, a sequence of functions fk : R → R such that (fk) (b) converges uniformly…

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Q: 4. If (1,) is Cauchy and has a convergent subsequnece (r), then prove that (r,) is convergent and…

A: Given that the sequence xn is a Cauchy sequence and it has a convergent subsequence xnk

Q: If |fn(x)| < an for all x and Ln=1 an converges, prove tha En-1 fn(x) converges uniformly. n=1

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

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Q: (e) $₁ = 5 and Sn+1 = √√4s +1 for n ≥ 1.

A: Given s1 = 5 and sn+1 = 4sn+1 for n≥1