Sum of Infinite Power Series

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 infinite series

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.