Set an where r E R is a real number. Prove that ɑn converges to 0 if (-1, 1), cônverges tó 1 if r = 1, and diverges otherwise.

Q: 28.P. Let f. (x)= |x]*1f for x in [-1, 1]. Show that each f. is differentiable on [-1, 1] and that…

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Q: Let xn -> 0 and (yn) be bounded. Prove xnyn -> 0.

A: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k,…

Q: Prove that if r >1 then f r" dx converges to 1

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Q: Find P (-1.5 < or equal to Z < 0.2) where Z ~ N (0,1)

A: We will use Standard Normal table to find the required probability.

Q: Let z(x) ~N(0,1) Prop (x > -0.56) = Prop (x ≤ -2.17) = Prop (-1.56 ≤ x ≤ 0.72) =

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Q: Prove that x² + ¹² +2111²2 = y 1112 where prime denotes differentiation w.r.t.s. 1 1+p¹² 4 p² o² + P

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Q: Prove that if f: A to R is continuous at x=c, then for all (x_n) converging to c, it follows that…

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Q: Use mathematical induction to prove the following statement derivatives of the function f(x) = x¯¹…

A: We have to prove that the nth derivative of  where  is using mathematical induction

Q: 2.66 Let fn(x) = z for all r E R. a) Find the pointwise limit of fn. b) Does f converge uniformly on…

A: Convergence and uniform convergence of the sequence

Q: Let Xi ∼ Bernoulli(1/2), and let Yn = X1 + X2 + · · · + Xn. Use the central limit theorem to find…

A: Xi ~Ber(1/2) ; i=1,2,...,n

Q: 2.J Let fn(x) = x ½n for 0<x< 1 and neN. or.) Find a function f on [at] such that (fr) converges.…

A: If each ,  is continuous and pointwise convergent on D and the sum function f of the sequence is…

Q: Ex filnd divergenceof f when yj at (o1). 2 e ( + 9 J

A: See the answer below

Q: -Prove that 2x² - 8x = 22 has at least one soution on the interval IVT. [5,10] by using

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Q: Prop (1.23 ≤ x ≤ 1.72) = Prop (-2.50 ≤ x ≤ 0.5) = Prop (x ≥ 3.00)

A: here use normal distribution table

Q: 99. Show that 00 cos x dx is absolutely convergent.

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Q: Give an example of a sequence of continuous functions: fn: [0, 1] R that is pointwise convergent but…

A: Consider the function fx=0, x is irrational in 0,1mn, x is rational in 0,1. This function is called…

Q: The graph of f is shown in the figure. 6. 4 х 14 -4 -6+ 2-

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Q: Use the MVT to prove that if f' (x) ≤ 2 for x &gt; 0 and f (0) = 4, then f (x) ≤ 2x + 4 for all x ≥…

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Q: f(x) = 3x - 2sınx +7 %3D aluse the intermediate Value Theore m to show that flx) has at least one…

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