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We define an arithmetic sequence of functions: fn: [1,0) → R recursively as: fi(x) = Vx, Vx € [1, 0) fn + 1 (x) = /x+ fn (x) ,Vx € [1,∞0) a- Prove that the sequence converges pointwise in [1, 0) and find its limit. b- Prove that for every a > 1 the sequence converges uniformly in [1, a]

We define an arithmetic sequence of functions: fn: [1,0) → R recursively as: fi(x) = Vx, Vx € [1, 0) fn + 1 (x) = /x+ fn (x) ,Vx € [1,∞0) a- Prove that the sequence converges pointwise in [1, 0) and find its limit. b- Prove that for every a > 1 the sequence converges uniformly in [1, a]

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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We define an arithmetic sequence of functions: fn: [1, 00) → R recursively as: f1(x) = Vx, fn + 1 (x) = /x + fn (x) ,Vx € [1,0) Vx E [1, 0) a- Prove that the sequence converges pointwise in [1, 00) and find its limit. b- Prove that for every a > 1 the sequence converges uniformly in [1, a]
Transcribed Image Text:We define an arithmetic sequence of functions: fn: [1, 00) → R recursively as: f1(x) = Vx, fn + 1 (x) = /x + fn (x) ,Vx € [1,0) Vx E [1, 0) a- Prove that the sequence converges pointwise in [1, 00) and find its limit. b- Prove that for every a > 1 the sequence converges uniformly in [1, a]
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