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Presentation Kim Do kyoung POSTECH May 1, 2008 Presentation - - PowerPoint PPT Presentation

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Presentation Presentation Kim Do kyoung POSTECH May 1, 2008 Presentation Introduction Math Analysis Suppose Then n f n ( x ) = 0 lim x [0 , 1] 1 lim f n ( x ) dx = 0 ? where f n : [0 , 1] R is n 0

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SLIDE 1

Presentation

Presentation

Kim Do kyoung

POSTECH

May 1, 2008

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Presentation Introduction Math

Analysis

Suppose lim

n→∞ fn(x) = 0

∀x ∈ [0, 1] where fn : [0, 1] → R is continuous ⇒ Then lim

n→∞

1 fn(x)dx = 0?

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Presentation Introduction Math

Analysis

Suppose lim

n→∞ fn(x) = 0

∀x ∈ [0, 1] where fn : [0, 1] → R is continuous ⇒ Then lim

n→∞

1 fn(x)dx = 0? The answer is

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Presentation Introduction Math

Analysis

Suppose lim

n→∞ fn(x) = 0

∀x ∈ [0, 1] where fn : [0, 1] → R is continuous ⇒ Then lim

n→∞

1 fn(x)dx = 0? The answer is, of course,

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Presentation Introduction Math

Analysis

Suppose lim

n→∞ fn(x) = 0

∀x ∈ [0, 1] where fn : [0, 1] → R is continuous ⇒ Then lim

n→∞

1 fn(x)dx = 0? The answer is, of course, No.

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Presentation Introduction Math

Counterexample Let fn(x) =    (n + 1)2(x −

1 n+1), 1 n+1 ≤ x ≤ 2 n+1

(n + 1)2(−x +

3 n+1), 2 n+1 ≤ x ≤ 3 n+1

0,

  • therwise

The following figure is when n = 4.

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Presentation Introduction American History

The Statue of Liberty

The tablet contains the text ”JULY IV MDCCLXXVI” (July 4, 1776) commemorating the date of the United States Declaration of Independence.

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Presentation Introduction American History

The Statue of Liberty

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