Let X = (xn) be a positive sequence of real numbers. Show that if x₁ + x₂ + + In Yn n then yn is always divergent (even if an converges).

Q: Let fn RR be a sequence of continuous functions which converg uniformly to a function f : R → R. Let…

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Q: 3 3) The Sequence of real function f(x) = x²+³ converges over a interval [1,1], dotally from th…

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Q: n=1 (ii) Give an example such that a, converges conditionally and diverges. IM:

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Q: 2ху lim (x.y)-(1.0) Which one of the following statements is correct? (a) Consider the limit OThe…

A: Each questions are solved with proper reasons.

Q: Show that sum of two uniformly convergent sequen ce is uniformly convergent

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A: This question is basically based on the theorem named Banach fixed point Theorem which states that:…

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

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Q: Show that f"(0) = 0. Use induction and the ideas from the above to show that f(k)(0) = 0 for every k…

A: Given that fx=∑n=0∞anxn in the interval -R,R. Then R is the radius of convergence of the function f.…

Q: 5. Consider the equation f(x) = x - 2- = 0. (a) Show that the above equation has a solution on the…

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Q: Show that >n"x" diverges for all x + 0. n=0

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Q: Determine of each point convergens or diverges: (-2)"(x-3)" (3) √(n+1) M=O

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Q: 8n Show that fn (x) = x2" – x" is unfiormly converges or not by using uniform norm

A: The given function is  fn(x)=x2n-x8n We show that the function is uniformly converges or not by…

Q: If xi 2, 3dn-1+ 3"n-2 show that {xn} is convergent. identify the limit.

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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…

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Q: A.) If {a,} is a convergent sequence, then {a,} is also a convergent sequence. B.) If f(x) is a…

A: This question is related to sequence and series.

Q: If an> 0 and Σ an converges, then Σ(-1)"an converges.

A: We have to check wheather it is true or not

Q: 1) In order to use the integral test, we must verify that the sequence is decreasing (and positive…

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A: Let the given sequence is Sn=-1n+1n We have to find two subsequences of given sequence that converge…

Q: e-Xdx converges xpx

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Q: 4. Suppose (xn) converges, and let K be a positive integer. Now create a sequence (yn) by changing…

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Q: Suppose xn2 0 for all n E N and the sequence (-1)"xn converges. What is its limit?

A: A sequence represented by say bn; if converges thenlimn→∞bn=0. It is important to note that the…

Q: value of for which Are there cmy value f converges.

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Q: (5) Let' (1) : r² + 7x + 12. S(z) dr (a) Use a comparison test to show that. A B converges. S(x) (b)…

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Q: Q(3) The sequence 4n3 3n7 + n3 2n7 (a) Converges to (b) Diverges (c) Converges to

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Q: 9) n+3"

A: Comparison test for series of positive terms states that, If an, bn are two series such that…

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A: Test for uniform convergence of sequences: Let fn be a sequence of functions such that it converges…

Q: k* for some constant k". Lp.) Show that L(po)

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

A: We have to prove that if convergence of (sn) implies the convergence of (|sn|).

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Q: The series 2n +2 Diverges to-o0 Converges to 1- Converges to B Diverges to +

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