Question

Let f_n(x) = nx(1 - x^2)^n be a sequence of functions that converges pointwise to f(x) on [0, 1]. (a) Calculate f(x) and ?_0^1 f(x). (b) Calculate lim_{n??} ?_0^1 f_n(x). (c) Does f_n converges uniformly on [0, 1]? Why? Calculate lim_{n??} ?_0^{1/2} sin(x^n). Hint: Use that the sequence f_n(x) = sin(x^n) converges uniformly on [0, 1/2].

          Let f_n(x) = nx(1 - x^2)^n be a sequence of functions that converges pointwise to f(x) on [0, 1].

(a) Calculate f(x) and ?_0^1 f(x).
(b) Calculate lim_{n??} ?_0^1 f_n(x).
(c) Does f_n converges uniformly on [0, 1]? Why?

Calculate lim_{n??} ?_0^{1/2} sin(x^n).
Hint: Use that the sequence f_n(x) = sin(x^n) converges uniformly on [0, 1/2].

Let fn(x) = nx(1 - x^2)^n be a sequence of functions that converges pointwise to f(x) on [0, 1].

(a) Calculate f(x) and ?0^1 f(x).
(b) Calculate limn?? ?0^1 fn(x).
(c) Does fn converges uniformly on [0, 1]? Why?

Calculate limn?? ?0^1/2 sin(x^n).
Hint: Use that the sequence fn(x) = sin(x^n) converges uniformly on [0, 1/2].

Added by Joseph O.

Calculus: Early Transcendentals

James Stewart 8th Edition

Chapter 11

Instant Answer

Solved by Expert Adi S

12/30/2023

Step 1

- For \(0 \leq x < 1\), the term \((1 - x^2)^n\) goes to 0 as \(n\) approaches infinity because \(0 \leq 1 - x^2 < 1\), and any number strictly between 0 and 1 raised to an infinite power approaches 0. - For \(x = 1\), \(f_n(1) = n \cdot 1 \cdot (1 - 1^2)^n = n Show more…

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Let f_n(x) = nx(1 - x^2)^n be a sequence of functions that converges pointwise to f(x) on [0, 1]. (a) Calculate f(x) and ∫_0^1 f(x). (b) Calculate lim_{n→∞} ∫_0^1 f_n(x). (c) Does f_n converges uniformly on [0, 1]? Why? Calculate lim_{n→∞} ∫_0^{1/2} sin(x^n). Hint: Use that the sequence f_n(x) = sin(x^n) converges uniformly on [0, 1/2].