# Limit of power series of discontinuous function

• Aug 27th 2011, 04:19 AM
parklover
Limit of power series of discontinuous function
Fn(x)=|Sin^n(X)| is convergent to F(x) where F(x) =1 when x=(m+1/2)pi and F(x) =0when x =otherwise

prove $\displaystyle \lim n\rightarrow \inf \int |F(x)-Fn(x)|dx \ne 0$ integrate from -infinity to +ve infinity

1. not sure how to integrate a function that is discontinuous at a single point
2 my main confusion is Lim n->inf Fn(x) = 0 which is equal to F(x). Intuitively I can tell the area under sin^n X is not zero even when n->infinity. but I am not sure how to prove it using mathematical formulae
• Aug 27th 2011, 05:11 AM
SammyS
Re: Limit of power series of discontinuous function
What have you tried?

Where are you stuck?
• Aug 27th 2011, 06:47 AM
CaptainBlack
Re: Limit of power series of discontinuous function
Quote:

Originally Posted by parklover
Fn(x)=|Sin^n(X)| is convergent to F(x) where F(x) =1 when x=(m+1/2)pi and F(x) =0when x =otherwise

prove $\displaystyle \lim n\rightarrow \inf \int |F(x)-Fn(x)|dx !=0$ integrate from -infinity to +ve infinity

1. not sure how to integrate a function that is discontinuous at a single point
2 my main confusion is Lim n->inf Fn(x) = 0 which is equal to F(x). Intuitively I can tell the area under sin^n X is not zero even when n->infinity. but I am not sure how to prove it using mathematical formulae

$\displaystyle F(x) \ne 0$ on a set of measure zero, so as far as integration is concerned you can replace it by zero.

CB
• Aug 27th 2011, 12:16 PM
parklover
Re: Limit of power series of discontinuous function
Quote:

Originally Posted by CaptainBlack
$\displaystyle F_n(x) \ne 0$ on a set of measure zero, so as far as integration is concerned you can replace it by zero.

CB

sorry ,not really understand. you mean set $\displaystyle F_n(x) = 0$ when n->infinity and integrate? How can you get end result not equal 0?
• Aug 27th 2011, 02:06 PM
CaptainBlack
Re: Limit of power series of discontinuous function
Quote:

Originally Posted by parklover
sorry ,not really understand. you mean set $\displaystyle F_n(x) = 0$ when n->infinity and integrate? How can you get end result not equal 0?

Sorry, typo I mean that $\displaystyle F(x)\ne 0$ on a set of measure zero, so:

$\displaystyle \lim_{n \to \infty}\int_{-\infty}^{\infty}|F(x)-F_n(x)|dx=\lim_{n \to \infty}\int_{-\infty}^{\infty}|F_n(x)|dx$

CB