THE MATH COLUMN
An Infinite Series Fallacy
By Dr. Alfred Posamentier
This is the time of year when we can provide some challenging entertainment – ones that can lead the motivated reader to do some further investigation. Here is one that will leave many readers somewhat baffled. Yet the “answer” is a bit subtle and may be require some more mature thought.
By ignoring the notion of a convergent series* we get the following dilemma:
Let S = 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + . . .
= (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + . . .
= 0 + 0 + 0 + 0 + . . .
= 0
However, were we to group this differently, we would get:
Let S = 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + . . .
= 1 – (1 – 1) – (1 – 1) – (1 – 1) – . . .
= 1 – 0 – 0 – 0 – . . .
= 1
Therefore, since in the first case, S = 1, and in the second case, S = 0, we could conclude that 1 = 0.
What’s wrong with this argument?
If this hasn’t upset you enough, consider the following argument:
Let S = 1 + 2 + 4 + 8 + 16 +32 + 64 + . . . . (1)
Here S is clearly positive.
Also, S – 1 = 2 + 4 + 8 + 16 +32 + 64 + . . . . (2)
Now by multiplying both sides of equation (1) by 2, we get:
2S = 2 + 4 + 8 + 16 + 32 + 64 + . . . (3)
Substituting equations (2) into (3) gives us:
2S = S – 1
From which we can conclude that S = – 1.
This would have us conclude that – 1 is positive, since we established earlier that S was positive.
To clarify the last fallacy, you might want to compare the following correct form of a convergent series:
Let S = 1 + (1/2) + (1/4) + (1/8) + (1/16)
We then have 2S = 2 + 1 + (1/2) + (1/4) + (1/8) + (1/16)
Then 2S = 2 + S, and S = 2, which is true. The difference lies in the notion of a convergent series as is this last one, while the earlier ones were not convergent, and therefore, do not allow for the assumptions we made. #
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* In simple terms, a series converges if it appears to be approaching a specific finite sum. For example, the series 1 + (1/2) + (1/4) + (1/8) + (1/16) + (1/32) + L converges to 2, while the series 1 + (1/2) + (1/3) + (1/4) + (1/5) + (1/6 + L does not converge to any finite sum but continues to grow indefinitely.
