2. Assume that Σak <∞, prove that are converges uniformly on [-1,1].

Q: Σ > lan| < o, (-1, 1] For prove that 2 anx" converges uniformly on n=0 EM:

A: The proof for the convergence on [-1, 1] is shown below.

Q: Problem 1: Consider the systems of equations below. For each system, write it as Ar = b, and then…

A: In this solution we will discuss the LU-decomposition of the corresponding coefficient matrices of…

Q: 2. Construct a sequence of unbounded functions In : [0,1] → R that converges pointwise to a bounded…

A: Step 1:

Q: For each nЄ Z+ let fn : R → R be the function пх fn(2) n|x| +1° (a) Prove that (fn) converges…

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Q: 7. Suppose A and B are compact subsets of R and that fn is a sequence of functions converging…

A:

Q: Suppose that ak ≥ 0 and that ak^1/3→ a as k → ∞.Prove that sigma,k from 1 to ∞,akx^k converges…

A:

Q: Suppose you run a limit comparison test and determine lanl diverges. In your written work: Nothing,…

A: If series of absolute term diverge than series doesnot converge absolutely.

Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

Q: Does the Bounded Convergence Theorem hold if m(E)<∞ but we drop the assumption that the sequence…

A: Given below clear explanation

Q: converges weakly to x in a normed linear sp

A: Given that, xn converges weekly to x in a normed linear space X We have to show that Every…

Q: 2. Suppose for every n ≥ 1 that {f} are twice differentiable functions on [a, b] such that both {f}…

A:

Q: 19) Prove or find a counter example to the following statements: a) f is continuous on (a,b) implies…

A: Step 1: Explanation for question

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

A: The objective of the question is to prove that a sequence converges if the absolute value of its…

Q: Let fn be a bounded sequence of functions uniformly convergetn to f. Prove that f is bounded as…

A:

Q: (-1)* k4/3 k=1 a) Converges both absolutely and conditionally b) Converges conditionally c)…

A:

Q: i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If SnX1 X2 + x3…

A: Given that, { xn } is a sequence of non negative terms and this sequence is monotonic decreasing,…

Q: 9. Prove that if {fn(x)} converges uniformly to the function f(x) on each of the sets A1, A2, ...,…

A:

Q: Exercise 1. A student is asked to prove If m is even and n is odd, then m² + n² is odd. 2 The…

A: Let us answer above questions

Q: Suppose that p(æ) > anx" converges on (-1, 1], find the internal of convergence of p(6x – 9). n=0

A: