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?
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:
2^{n} +
2^{n1} +
2^{n2} +
... + 2^{3} +
2^{2} +
2^{1}^{ }=
2^{n+1} 
2 = 2(2^{n} 
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:
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.
