Homework_Set_9_Solution

Homework_Set_9_Solut - Math 1A Fall 2005 HOMEWORK 9 SOLUTIONS 1 Suppose f is a continuous function on[0 ∞ so that sup x Z x | f y | dy< ∞

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Unformatted text preview: Math 1A Fall 2005 HOMEWORK 9 SOLUTIONS 1. Suppose f is a continuous function on [0 , ∞ ) so that sup x Z x | f ( y ) | dy < ∞ Prove that R ∞ f ( y ) dy exists. ( Hint: Look at our proof of the analogous fact for series using | a n | + a n ; see Theorem 10.15 in Apostol.) Solution. We define R ∞ f ( y ) dy as Z ∞ f ( y ) dy = lim x →∞ Z x f ( y ) dy. Since f ( y ) is continuous, this implies that | f ( y ) | + f ( y ) is continuous, and so is integrable on [0 , x ]. Note that Z x f ( y ) dy = Z x ( | f ( y ) | + f ( y )) dy- Z x | f ( y ) | dy. If we can show that lim x →∞ R x ( | f ( y ) | + f ( y )) dy and lim x →∞ R x | f ( y ) | dy exist, then we will have shown by limit laws that R ∞ f ( y ) dy exists. • To see that lim x →∞ R x | f ( y ) | dy exists, let L = sup x Z x | f ( y ) | dy. Now by definition of supremum, for any ε > 0 there exists M ≥ 0 such that Z M | f ( y ) | dy > L- ε. (If this were not true, then L- ε would be an upper bound for each R M | f ( y ) | dy , and L would no longer be the least upper bound.) If x ≥ M , then we have Z x | f ( y ) | dy = Z M | f ( y ) | dy + Z x M | f ( y ) | dy ≥ Z M | f ( y ) | dy > L- ε, where we have used the fact that | f ( y ) | ≥ 0 = ⇒ R x M | f ( y ) | dy ≥ 0. Thus we have shown that for any ε > 0 there exists M ≥ 0 such that, whenever x ≥ M , we have Z x | f ( y ) | dy- L = L- Z x | f ( y ) | dy < ε. But this precisely means that lim x →∞ R x | f ( y ) | dy = L . 1 • Next, observe first that | f ( y ) | + f ( y ) = 2 f ( y ) if f ( y ) ≥ 0 and | f ( y ) | + f ( y ) = 0 if f ( y ) < 0; hence | f ( y ) | + f ( y ) ≥ 0. Moreover, f ( y...
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This note was uploaded on 03/30/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.

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Homework_Set_9_Solut - Math 1A Fall 2005 HOMEWORK 9 SOLUTIONS 1 Suppose f is a continuous function on[0 ∞ so that sup x Z x | f y | dy< ∞

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