Consider the infinite series S

S = 1/√(1) + 1/√(2) + 1/√(3) + 1/√(4) + 1/√(5) + 1/√(6) + 1/√(7) + ......

Jacob Bernoulli knew that this sum of reciprocals of square roots, must

diverge since the denominators were all smaller than the harmonic series.

Let the inf sum of odd terms, O = 1/√(1) + 1/√(3) + 1/√(5) + 1/√(7) + .....

and the inf sum of even terms, E = 1/√(2) + 1/√(4) + 1/√(6) + 1/√(8) + ...

By inspection O + E = S, but we may also write √(2)E = S, so we have

O + E = √(2)E, or,

O = [√(2) – 1]E  ~ 0.4142E

Bernoulli remarked on the apparent paradox that the odd sum seems less than the even sum, but this is impossible because term by term odd terms are larger

Can you resolve the paradox with a clear explanation?

Relevance

In Bernoulli's and his contemporaries' manipulation with infinite series (not just this one), there are two remarkable facts:

a) The assignation S=∑ 1/√n

b) The procedure consisting of obtaining new series (and consequently, new results) from adding, subtracting and replacing expressions by known series.

For us, problem would be point a), that is to say, the fact that Bernoulli refer to the series ∑ 1/√n as if it had a real value assigned to it.

If this difficulty is overlooked, point b) may be justified by the rearrangement theorem as the adding and subtracting of series with positive terms is a common procedure when the series involved are convergent.

Since the series ∑ 1/√n is divergent, to assign a value to it is as absurd as pretending that S = 1 + 2 + 3 + 4 + · · · is a determined quantity.

Since we cannot write S = ∑ 1/√n, we are forced to consider only the partial sums of the series Sₙ = ∑1/√n

We may sum first two terms of S

S₂= 1/√1 + 1/√2 = 1 + 1/√2

There is one even term in it, E₁=1/√2

Quotient between the sums S₂/E₁ = (1+1/√2 )/(1/√2) = 1+√2 ≈ 2.414

is significantly different from that in your question

And if we take 4 terms

S₄= 1/√1 + 1/√2 + 1/√3 + 1/√4 = 1/√3 + (3+√2)/2

There are two even terms, E₂=1/√2 +1/√4 = (√2+1)/2

S₄/E₂ = (2√6 - 2√3 + 6√2 - 3)/3 ≈ 2.307

And (without proof)

lim n⭢∞ S₂ₙ/Eₙ = 2

• Anonymous
4 months ago

Because the series diverge O,E,S are not real numbers and hence you cannot perform arithmetic operations between them. If you could things like "∞+1=∞ so 1=0" would be valid.