Puzzling Visual Maths For The Curious Minded #3  
     
  archimedes lab logo
Archimedes Laboratory Project

Mental & Perceptual Activities that Enhance Critical and Creative Thinking Skills

Social Media Contact us Like us on Facebook Circle us on Google+ Follow us on Twitter Pin it Thumb up this page

 

 

Puzzling Visual Maths

Button Backward

Sum of Infinite Power Series Button Forward
 

Have a look at the two distinct sums of series of powers below.

Same procedure, different result accuracy levels... Can you guess what went wrong in the operation of fig. 2?

Sum of infinite powers

Some of you may be puzzled by the paradoxical result of the operations in fig. 2, in fact: infinity ≠ -2.

Moreover, you can find in any math handbook that the sum of powers of 2 gives:

2n + 2n-1 + 2n-2 + ... + 23 + 22 + 21 = 2n+1 - 2 = 2(2n - 1)

So, were is the error?

The Math Behind the Fact: The Indetermination of ∞ - ∞
While the limit of the sum of fractions can converge to a limit, in this specific case to 1, the sum of powers doesn’t have a limit because it cannot exist since:

lim Sn = ∞
n → ∞  

So, you cannot subtract S from both sides of the equation; because that would be writing:

- 2 + ∞ - ∞ = 2∞ - ∞

and the problem is that even in the extended reals*, ∞-∞ is undetermined. It does not equal anything, and certainly not zero. In short, you cannot just cancel infinities.

*In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and -∞ (read as positive infinity and negative infinity respectively). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. Source Wikipedia.

 
 

 

You can comment and discuss this puzzle on our FaceBook fan page.

 

From the same Author
Impossible Folding Puzzles
Impossible Folding Puzzles

Math products
Postcards
Postcards
T-shirt man
T-shirt man
T-shirt girl
T-shirt girl
 
     
 
 

archimedes lab logoArchimedes Laboratory Project
Mental & Perceptual Activities that Enhance
Critical and Creative Thinking Skills

Social Media Contact us Like us on Facebook Circle us on Google+ Follow us on Twitter Pin it Thumb up this page
arrow 1ABOUT arrow 1SITEMAP arrow 2 SERVICES arrow 2 PRODUCTS arrow 1SUBMIT arrow 1ADVERTISE arrow 1CONTACT
 

 

   For Publishers
   Workshops
  Books
  Puzzles
  Prints & Posters
  Illusory Art
     
Archimedes in the bathtub
 
Contact us Like us on Facebook Circle us on Google+ Follow us on Twitter Pin it Thumb up this page
 
  © 1997-2014 G. Sarcone [Archimedes-lab.org] All rights reserved  
  Creative Commons LicenseArchimedes Laboratory™ by Gianni A. Sarcone is licensed under a
Creative Commons Attribution - NonCommercial - NoDerivatives 4.0 International License