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Bernoulli's Paradox

Bernoulli's paradox is the apparent contradiction obtained by assuming that the harmonic series has a finite sum h and then deriving h-1=h (Boas 1979, Dunham 1987). It provides a proof by contradiction that the harmonic series diverges.

Suppose that

sum_(k=1)^infty1/k=h.
(1)

For each positive integer k, the telescoping series identity

sum_(j=k)^infty1/(j(j+1))=sum_(j=k)^infty(1/j-1/(j+1))=1/k
(2)

expresses 1/k as a sum of positive terms. Substituting these expansions gives a double series. Under the assumption that h is finite, its positive terms may be regrouped by j without changing the sum. For a fixed j, the term 1/[j(j+1)] occurs once for each of k=1, ..., j, hence exactly j times. Therefore,

h=sum_(k=1)^(infty)sum_(j=k)^(infty)1/(j(j+1))
(3)
=sum_(j=1)^(infty)j/(j(j+1))
(4)
=sum_(j=1)^(infty)1/(j+1)
(5)
=h-1.
(6)

This gives h-1=h, which is impossible for finite h. The harmonic series therefore diverges (Dunham 1987, pp. 19-21).

See also

Harmonic Series, Telescoping Series

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References

Boas, R. P. "Some Remarkable Sequences of Integers." Ch. 3 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 39-40, 1979.Dunham, W. "The Bernoullis and the Harmonic Series." College Math. J. 18, 18-23, 1987. https://doi.org/10.1080/07468342.1987.11973001.

Referenced on Wolfram|Alpha

Bernoulli's Paradox

Cite this as:

Weisstein, Eric W. "Bernoulli's Paradox." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BernoullisParadox.html

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