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:

2*S* = 2 + 4 + 8 + 16 + 32 + 64 + . . . (3)

Substituting equations (2) into (3) gives us:

2*S = 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 2*S* = 2 + 1 + (1/2) + (1/4) + (1/8) + (1/16)

Then 2*S* = 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.