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) Suppose fn is a sequence of continuous functions on [0, 1] which are continuously differentiable on (0, 1) such that E
Posted: Thu Apr 28, 2022 6:33 am
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- Suppose Fn Is A Sequence Of Continuous Functions On 0 1 Which Are Continuously Differentiable On 0 1 Such That E 1 (116.59 KiB) Viewed 53 times
) Suppose fn is a sequence of continuous functions on [0, 1] which are continuously differentiable on (0, 1) such that En=1 f'n converges uniformly to a bounded function on (0, 1). Also, fn(0) converges. Prove that 2n=1 fn converges to a continuous function on [0, 1], which is differentiable on (0, 1). a Use the Integral Test Theorem for proof. Integral Test Theorem 6.2.1. Suppose f is a positive, non-increasing function on [1, 0o) and f(k) for each k € N. Then the series ak converges if and only if the ~ ak k=1 improper integral 5° f(x) dx converges. , .