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Prove that En-1 (nt) converges uniformly on S = [0, 1]. x2n =1 (n+x)²

Prove that En-1 (nt) converges uniformly on S = [0, 1]. x2n =1 (n+x)²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Real Analysis 2 Please follow similar sample in 2nd photo to solve problem
x ex & fn(x) = x +n for all x 20 (s = [0, 0)) Show for f phoise. S where f =0 a. b. Show frf Unif c. Show for * > f Chrif. Solna @tix XES = x = x+n Notice fn (x) = x+n, So (in fri (x) 130 goal on = xtr #2>0: X+0 on [0₁2] b. Show for f urif. on > By AP. I NE N sit / x ¾/2 So 2+N </N LE· Fix X € [0₁2]. L X 스 x+n on S. [0, ∞). Show fr (x) → 2 Then X+n = 2+ (So fn(x) = fm (2)) Assume n>x. Then (fn (x) = f(x)) = /*+₁ -0 2¹ LE 스 2f0 2+1 | fn(x) = f(x) | < E - -02E २६ Cam Xnº n->∞ x+n' f [0,2]. fix E70 fo is mor (>N) if n is large enough SE 2 & 2² ≤ ¾N CE 스 2+n 2+1
Transcribed Image Text:x ex & fn(x) = x +n for all x 20 (s = [0, 0)) Show for f phoise. S where f =0 a. b. Show frf Unif c. Show for * > f Chrif. Solna @tix XES = x = x+n Notice fn (x) = x+n, So (in fri (x) 130 goal on = xtr #2>0: X+0 on [0₁2] b. Show for f urif. on > By AP. I NE N sit / x ¾/2 So 2+N </N LE· Fix X € [0₁2]. L X 스 x+n on S. [0, ∞). Show fr (x) → 2 Then X+n = 2+ (So fn(x) = fm (2)) Assume n>x. Then (fn (x) = f(x)) = /*+₁ -0 2¹ LE 스 2f0 2+1 | fn(x) = f(x) | < E - -02E २६ Cam Xnº n->∞ x+n' f [0,2]. fix E70 fo is mor (>N) if n is large enough SE 2 & 2² ≤ ¾N CE 스 2+n 2+1
x2n 1. Prove that Σn-1 (n) converges uniformly on S = [0, 1].
Transcribed Image Text:x2n 1. Prove that Σn-1 (n) converges uniformly on S = [0, 1].
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Recall the M-test for uniform convergence.

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