Ian H
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Ian H asked in Science & MathematicsMathematics · 4 months ago

Series paradox question?

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? 

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  • 4 months ago
    Favourite answer

    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.

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