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< ∞

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 5

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< ∞

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online