Find the flaw: something = nothing

[BMAD]

Something = nothing?

Let S= 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + . . . . . . .

Then taking 1/2 of each term

1/2 S= 1/4 + 1/6 + 1/8 + 1/10 + 1/12 + 1/14 + . . . . . . .

Subtract 1/2 S from S, the result is

1/2 S= 1/2 + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + . . . . . . . . (By removing 3rd, 5th, 7th, and so on from the original terms set)

Subtracting the first expression for

1/2 S from the second expression for

1/2 S, we get

1/2 + (1/3 - 1/4) + (1/5 - 1/6) + . . . . = 0

Or 1/2 + 1/12 + 1/30 + 1/56 + 1/90 = 0

Proving that something equals nothing.

Edited January 3, 2015 by BMAD

[DejMar]

When taking half the terms of S, the rate at which the infinite series of terms approached 0 was doubled. In addition, the first term of one series of terms, i.e., 1/2, is also set aside before matching up the terms for subtracting one expression from the other, as if the infinite series of the half terms had one more term than the infinite series of the original expression.

Consider the series as finite:
S = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7.
1/2 S = 1/4 + 1/6 + 1/8 + 1/10 + 1/12 + 1/14
Subtracting 1/2 S from S
1/2 S = 1/2 + 1/3 + 1/5 + 1/7 + (-1/8 -1/10 -1/12 -1/14)
Note, there are subtractive terms that have been failed to be expressed by the problem's falacial logic. In the infinite series where one 1/2 S expressions is subtracted from the other, these subtractive terms will total to the negative equivalence of the sum of the addititive terms, thus the the left-hand side of the expression is not the falacial something, but also is nothing (zero).

Edited January 3, 2015 by DejMar

[bonanova]

S is the harmonic series, which diverges.

You're playing with infinity, which can lead to trouble.

 

Take the sum of all the integers. Call it S

Take the sum of every other integer (say the even ones.) That sum is S/2.

Because S is infinite, S/2=S, so their difference is zero.

 

Their difference is the sum of the odd integers.

The sum of the odd integers is zero.