Daily Maths Challenges

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Infinite product follow-up (links.hackliberty.org)

submitted 3 months ago by [email protected] to c/[email protected]

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This is related to the the May 16 post, but takes only the prime indexed terms. Does it still diverge?

Hint

Transform the product into a sum

Hint

The harmonic series 1 + 1/2 + 1/3 + ... 1/n +... diverges

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[–] [email protected] 6 points 3 months ago* (last edited 3 months ago) (1 children)

I don't know how to begin proving it, but the more I run this series out, bigger it gets. The conditions of the equation are such that it will always have a consistently non-zero rate of increase, even though that rate of increase decreases each time the formula is cycled ((p~n~/p~n~-1) will always be more than (p~n+1~/p~n+1~-1), nonetheless any and every (p~n~/p~n~-1) will be >1). The divergence will be glacial, but definite.

[–] [email protected] 4 points 3 months ago

I can confirm that your intuition for divergence is correct.