Sums of infinitely many fractions: 1

Page last updated 02 Jul 2021

Consider the case of a pattern of a sum of positive numbers such as:

¹⁄₁₀ + ¹⁄₁₀₀ + ¹⁄₁₀₀₀ + ¹⁄₁₀₀₀₀ + ¹⁄₁₀₀₀₀₀ + …

In such a series, where all the numbers are positive, when calculating the limiting value of the series, it makes absolutely no difference as to the order in which the numbers come in the series. For example, the limit of the sum of the series:

¹⁄₁₀ + ¹⁄₁₀₀ + ¹⁄₁₀₀₀ + ¹⁄₁₀₀₀₀ + ¹⁄₁₀₀₀₀₀ + ¹⁄_1000000 + ¹⁄_10000000 + ¹⁄_100000000 + ¹⁄_1000000000 + ¹⁄_10000000000 + ¹⁄_{1000000000000000} …

is precisely the same as the limit of the sum of the series:

¹⁄₁₀₀₀₀₀ + ¹⁄₁₀₀₀₀ + ¹⁄₁₀₀₀ + ¹⁄₁₀₀ + ¹⁄₁₀ + ¹⁄_10000000000 + ¹⁄_1000000000 + ¹⁄_100000000 + ¹⁄_10000000 + ¹⁄_1000000 + ¹⁄_{1000000000000000} …

where, in the second case, each group of five numbers has the sequence reversed.

But we can also have patterns that include negative numbers. Such as:

(a)1 − ¹⁄₂ + ¹⁄₃ − ¹⁄₄ + ¹⁄₅ − ¹⁄₆ + ¹⁄₇ − ¹⁄₈ + …

This is a series that has been the subject of much study, and it is called the ‘Alternating Harmonic Series’. And it is a series that has a most remarkable property – when it comes to calculating the limit of the sum of the series, it does matter in which order the terms of the series are added. So, for example, the series:

(b)1 − ¹⁄₂ − ¹⁄₄ + ¹⁄₃ − ¹⁄₆ − ¹⁄₈ + ¹⁄₅ − ¹⁄₁₀ − ¹⁄₁₂ + ¹⁄₇ − ¹⁄₁₄ − ¹⁄₁₆ + …

which contains precisely the same terms as the series (a) above, but in a different order, has a limiting value of exactly half of the limiting value of the sum of the series (a). (Footnote: This applies to the harmonic series shown here, but different limiting summations can also apply to other series of positive and negative terms, where the terms occur in different orders (provided the series satisfies certain conditions). See, for example Wikipedia: Alternating Series.)

This remarkable fact regarding the order of these series was proved by the distinguished mathematician, Bernhard Riemann, about 160 years ago. He proved that you can change the order of the terms of the series to give practically any value you want for the limiting sum of the series.

This is truly astonishing because here we have a result that has been known and well-established for many years without controversy, yet it is a result that is utterly irreconcilable with the conventional Platonist viewpoint. The conventional Platonist viewpoint is that all numbers ‘exist’ and that sums of limitlessly many numbers, such as in the alternating harmonic series can ‘exist’ (For more on Platonism see Platonism, The Myths of Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.)

But how can a sum of numbers ‘exist’, but at the same time, the sum has infinitely many different values depending on the order in which the numbers are added?

The answer is simple – it can’t ‘exist’, since it is a fundamental property of addition that the order of addition is immaterial. (Footnote: The commutative property of addition.) This is another demonstration that incremental additions can never somehow magically change a finite quantity into a limitlessly large quantity of entities (see Platonism and the Infinite). As in the case of the Courant & Robbins contradiction, the Alternating Harmonic Series demonstrates that the Platonist viewpoint that the sum of a limitless quantity of fractions ‘exists’ is logically untenable.

And more …

For more demonstrations of contradictions arising from the Platonist beliefs in the ‘existence’ of ‘actual’ infinite sets, see The Courant & Robbins contradiction and Sums of infinitely many fractions: 2.

The page Lebesgue measure theory also demonstrates in detail why the conventional assumption that you can add the lengths of infinitely many intervals as an infinite sum is naive and simplistic, and overlooks a crucial fact.

You can also see a formal paper on some of the problems of calculating the measure of some sets that are defined in terms of limitlessness, see PDF On Smith-Volterra-Cantor sets and their measure.