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Define the Function and Determine the Pointwise Limit

For each natural number n, define the function g_n: [0,1] --> R by (g_n)(x) = nx(1- x^2)^n.

a. Determine the pointwise limit of the sequence {g_n}. Be sure to prove that your claim for the pointwise limit is correct.

b. Does the sequence {g_n} converge uniformly on {0,1]? Explain.

c. Evaluate and compare the two limits of g_n. You are welcome to use the fundamental theorem of calculus and/or ideas about area to evaluate the integrals.

A Word document is attached which presents the question under proper format.


Solution Summary

This solution is presented within an attached Word document and explains the three proofs in a detailed, step by step manner.