Question

for these 2 theorems, pick on hypothesis, remove it, and provide a counterexample showing that the new statement is false.

(d) Thm The Weierstrass Uniform Convergence Criterion): The sequence of functions converges uniformly to some f: D R iff the sequence In is uniformly Cauchy ,, : D (c) Thm (Differentiation of Power Series): If a power series converges on (-r,r) then it has all derivatives there and those derivatives may be found by differentiating the power series term-by- term.

Answer 1

Pooo uniformly Cauchy on f) onvenzel

0n l Lohere d Inlo.we have 14 3 1 Cnoose N- 0 on R

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LuTh may be dan m:o Contered at c colth ndrus Conwegente re The pouder Sesnies vepuesents a Continou tsattdesentiated team-by The poeoen da m-1 Scanned with amscanner

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