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Intro to Analysis in-class_Part_55

# Intro to Analysis in-class_Part_55 - 7.3 Uniform...

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7.3 Uniform convergence 115 Theorem 7.3.15 ((Baby) dominated convergence thm) . Suppose f n ∈ R [ a, b ] for every 0 < a < b < , and suppose f n unif ----→ f on every compact subset of (0 , ) . If g R [0 , ) , then | f n | ≤ g = lim n →∞ Z 0 f n ( x ) dx = Z 0 f ( x ) dx. Proof. HW. Theorem 7.3.16 (Stirling’s Formula) . lim x →∞ Γ( x +1) ( x/e ) x 2 πx = 1 . Proof. HW. Often: lim n →∞ n ! ( n/e ) n 2 πn = 1, meaning that n ! ( n/e ) n 2 πn . 7.3.6 Term-by-term differentiation Example 7.3.11. Recall f n ( x ) = sin nx n . This is uniformly dominated by 1 n , so converges uniformly to f 0, but f 0 n ( x ) 9 f 0 ( x )! Not even uniform convergence can save us now! Need stronger hypothesis. Theorem 7.3.17. Let f n C 1 ( I ) , f n pw ---→ f and f 0 n unif ----→ g . Then f C 1 ( I ) and f 0 ( x ) = g ( x ) . Proof. Fix a point a I . Then FToC1 gives f n ( x ) - f n ( a ) = Z x a f 0 n ( t ) dt n →∞ -----→ Z x a g ( t ) dt. However, we also have f n ( x ) - f n ( a ) f ( x ) - f ( a ), so apply FToC2 to f ( x ) - f ( a ) = Z x a g ( t ) dt to see that f C 1 ( I ) with f 0 ( x ) = g ( x ).

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Intro to Analysis in-class_Part_55 - 7.3 Uniform...

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