# Advanced Math Problems

## Transcribed Text

1. For n ∈ N, set fn(x) = x n/n, x ∈ [0, 1]. Prove that the sequence {fn} converges uniformly to f(x) = 0 on [0, 1]; that the sequence {f 0 n (x)} converges pointwise on [0, 1], but that {f 0 n (1)} does not converge to f 0 (1). 6. Show that each of the following series converge on the indicated interval and that the derivative of the sum can be obtained by term-by-term dierentiation of the series. a. P∞ k=1 1 (1+kx) 2 , x ∈ (0,∞) b. P∞ k=1 e −kx, x ∈ (0,∞) c. P∞ k=0 x k , |x| < 1 d. P∞ k=0 x k k! , x ∈ (−∞, ∞)

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