Question

Use the theorem on uniform convergence and integration to prove that fn​ (x)=n²x(1−x² )ⁿ converges pointwise but not uniformly to the zero function on [0,1].

Answer 1

To prove that fn​ (x)=n²x(1−x²)ⁿ converges pointwise but not uniformly to the zero function on [0,1], use the theorem on uniform convergence and integration. Find the pointwise limit as n approaches infinity, which is 0 for any x in the interval. Assume uniform convergence and reach a contradiction using integration. The function fn​ (x) converges pointwise but not uniformly to the zero function on [0,1].

Step-by-step explanation:

To prove that the function fn​ (x) = n²x(1−x²)ⁿ converges pointwise but not uniformly to the zero function on [0,1], we can use the theorem on uniform convergence and integration.

1. Start by finding the pointwise limit of fn​ (x) as n approaches infinity. The limit of n²x(1−x²)ⁿ as n approaches infinity is 0 for any x in the interval [0,1]. Therefore, the function converges pointwise to the zero function on [0,1].
2. To show that the function does not converge uniformly, assume the contrary that it does. Then, we can use integration to reach a contradiction. Integrate the absolute value of fn​ (x) from 0 to 1.
3. The integral of |fn​ (x)| from 0 to 1 is equal to n⁻²/n⁻ⁿ = n²/(n+1) which approaches 1 as n approaches infinity. Since the integral does not converge to 0, the function does not converge uniformly to the zero function on [0,1]. Therefore, the function fn​ (x) converges pointwise but not uniformly to the zero function on [0,1].