All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math > Calculus
Step Function (Posted on 2004-07-07) Difficulty: 3 of 5
A step function f(x) is defined on the interval [0, 1)

From [0, 1/2), the step is 1/2 high.
From [1/2, 3/4), the step is 3/4 high
From [3/4, 7/8), the step is 7/8 high
From [ 1 - 1/(2^k), 1 - 1/(2^(k+1)) ) the step is 1 - 1/(2^(k+1)) high

What is the exact area under the step function?

See The Solution Submitted by Brian Smith    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution | Comment 6 of 10 |

I'm a little late to come in, and I haven't peeked yet.

The area is a sumation of an infinite sequence:

.5^1*(1-.5^1)+
.5^2*(1-.5^2)+
.5^3*(1-.5^3)+...

Each term is .5^n*(1-.5^n).
Simplify to 2^-n - 2^-2n.

The infinite sum of such a sequence would seem to equal the sum of all 2^-n where n is positive and odd.  So the first term is 1/2 and every next term is a fourth of the previous.  In binary, the decimal would be .101010..., which is equal to 2/3.


  Posted by Tristan on 2004-07-08 15:38:40
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (11)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information