7. Prove that a converges if kak→ 0 as k→ ∞ for some > 1. Does the result hold if a = 1?

Q: 11. If {an} and {b„} are sequences of real numbers, lim,0 an = 5, lim, bn = 8, and fn(x) = a, + bnx,…

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Q: 21. Let (gn) be a sequence of Riemann integrable functions from [0, 1] into R such that |gn (x)| ≤ 1…

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Q: 5a) When p is a real number greater than 1, it can be shown that the improper 1 1 integral |dx…

A: Convergence of Improper Integrals at left-end point: Let us suppose that x=a be the point of…

Q: ¹0 1 √x + x4 -dx converges.

A: Comparison test for improper integrals: i If fx≤gx and ∫abgx dx converges, then ∫abfx dx also…

Q: 24.1 Let fn(x) = nx 1+2 cos² nr. Prove carefully that (fn) converges √√n uniformly to 0 on R.

A: We have to show that  converges uniformly to  on

Q: 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a?…

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Q: 3. For each n€ Z+ let fn : [0, 1] → R, fn(x) = 1/(1+x"). (a) Prove that (fn) converges uniformly.…

A: We have solved both the parts. I hope it will help you.Explanation:### Part (a): Prove that (fn​)…

Q: (n) Q(n) Show that, if P(x) and Q(x) are polynomials of positive degrees, then the sequence an =…

A: Given: Px and Qx are polynomials of positive degree. To show:The sequence an=PnQnn converges to 1.

Q: 3. Prove that the series 1 x2n n! 1 + x²n n=0 converges uniformly on R to a continuously…

A: We'll follow these steps:Show that the series converges uniformly on R.Prove that the limit function…

Q: Show that Show that Show that fexx dx converges for any p = R. In xdx and f′ (x(ln x)(1 + x²)−¹)dx…

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Q: 2) Suppose that f(x) = 5x² over the interval [1, 2]. Let Ax = 1/n. Let x₁=1+iAx. Find the limit of…

A: Given: fx=5-x2.Formula to be used: ∑i=1ni=nn+12 ; ∑i=1ni2=nn+12n+16

Q: Let fn(x) = sin" x and let 8 > 0. a) Prove: fn converges uniformly on [0, 5 – 8], but not uniformly…

A: The given function is fnx=sinnx. To find: (a) fnx=sinnx uniformly or not on given domains.…

Q: 27. Show that n=1 n! nn converges. Hint: Use lim n→∞ n (₁ + ²)² = n e.

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Q: Show by example that Σ(an/bn) may converge to something other than A/B even when A = gan, B = Σbn ≠…

A: Given that an, bn≠0 Let us consider ∑an converges to A. Also ∑bn converges to B. To show that ∑anbn…

Q: Show that if X ~ Poisson(A), then there exists m ER such that pX (k) is increasing for ksm, and…

A: A discrete random variable X has a Poisson distribution, with parameter 0}">{\displaystyle…

Q: Suppose f(x) > 0 and g(x) > 0 and that | f(x)dx diverges. If f(x) > g(x), what can we conclude about…

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Q: 2.62 Show that fn(x) = x+n converges uniformly on [0,1] but not on (0, 0).

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Q: Find the Taylor series for f (x) = sin 2x at a = n and its radius of convergence. Show all work.

A: Given function, fx=sin2x Point, a=π Taylor series of a function f(x) at point a is given by…

Q: Exercise 2. Determine whether the sequence of functions (fn)n converges uniformly on S. 1 fn(x) = S…

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Let fn(x) = x − x”. Prove that fn converges pointwise on [0,1] an find the limit function. Is the…

A: given sequence of function fn(x)=x-xn on [0,1]f(x)=lim fn(x)n→∞when 0≤x<1, lim fn(x)=x-0=x=f(x)…

Q: 5. Let yı := n √p, where p > 0, and yn+1 := √p+y, for nЄ N. Show that (y) converges and find the…

A: