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THEOREM 205. Define the functions fr : [0, 1] + R by Sn(:1) = x" /n...

THEOREM 205. Define the functions fr : [0, 1] + R by Sn(:1) = x /n for n E N. The sequence n H Sn converges uniformly to the

THEOREM 205. Define the functions fr : [0, 1] + R by Sn(:1) = x" /n for n E N. The sequence n H Sn converges uniformly to the function f = 0, but the sequence n o fh does not converge to f' = 0. Note that the operations of taking a limit and taking a derivative do not necessarily commute.
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Theorem? (205) - fritat ini [0, 1] - IR 1 fn (1) = 2 for nein. het Uniform convergence het SCR and for each 1 NEN; let fnis Itot This proves that in converges uniformly to the function fro.. Boved I method: Mn-test: s pointurise het Lefn} converges pdin) = xh put drain) LE 7 xhl LE log on-l <loge = (h-1) loga < loge >> - (n-1) logn > -dogt *- (n-1) dog var > loge lng (2) (

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