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A while back I posted a question about this series on the General Math forum and was brought to task for not showing any math.

My hope is to prove that these series are infinite.

http://oeis.org/A002378" [Broken] are the series 0,2,6,12,20,30... and distances between consecutive numbers are increasing even spans. The mid-points of those spans are consecutive squares.

This is the main series where we combine the low half-pronic span with the high half-pronic span for each step.

[PLAIN]http://math.rudytoody.us/LogVersionX.png [Broken]

x is: 1.500989... which equals y + z

This is a convergent series that involves only the high half-pronic spans.

[PLAIN]http://math.rudytoody.us/LogVersionY.png [Broken]

y is: 0.851916...

This is a convergent series that involves only the low half-pronic spans.

[PLAIN]http://math.rudytoody.us/LogVersionZ.png [Broken]

z is: 0.649073...

It is obvious that if a span was prime-free, these sums would be greater than 1 and the series would diverge.

This is what I call the Bertrand Series, based on Bertrand's Postulate, which states that n < p <= 2n and has been proved several times. I use 2^n < p <= 2^n+1 to create the series.

[PLAIN]http://math.rudytoody.us/LogVersionW.png [Broken]

w is: 0.868897...

Because it it known that there are one or more primes between each power of two, no exception can cause the series to diverge.

Plotting the first dozen steps of each series shows a line up for 2 steps and then an abrupt right turn into a horizontal line. This is what makes me think they converge.

Because each step produces a square-free denominator whose factors are the primes in that span, every step will be co-prime to all other steps and to the denominator of the sum. The numbers can be summed in any order. Because the denominators are made up of the product of the primes for each span, occasionally step n will be less than step n+1. But, the overall trend is decreasing.

I have validated each sum to over 500,000 unchanging digits after about 10-20 steps.

My questions are:

1) What is required to prove convergence?

2) What would it take to prove that a prime-free span cannot occur for the pronic series?

My hope is to prove that these series are infinite.

http://oeis.org/A002378" [Broken] are the series 0,2,6,12,20,30... and distances between consecutive numbers are increasing even spans. The mid-points of those spans are consecutive squares.

This is the main series where we combine the low half-pronic span with the high half-pronic span for each step.

[PLAIN]http://math.rudytoody.us/LogVersionX.png [Broken]

x is: 1.500989... which equals y + z

This is a convergent series that involves only the high half-pronic spans.

[PLAIN]http://math.rudytoody.us/LogVersionY.png [Broken]

y is: 0.851916...

This is a convergent series that involves only the low half-pronic spans.

[PLAIN]http://math.rudytoody.us/LogVersionZ.png [Broken]

z is: 0.649073...

It is obvious that if a span was prime-free, these sums would be greater than 1 and the series would diverge.

This is what I call the Bertrand Series, based on Bertrand's Postulate, which states that n < p <= 2n and has been proved several times. I use 2^n < p <= 2^n+1 to create the series.

[PLAIN]http://math.rudytoody.us/LogVersionW.png [Broken]

w is: 0.868897...

Because it it known that there are one or more primes between each power of two, no exception can cause the series to diverge.

Plotting the first dozen steps of each series shows a line up for 2 steps and then an abrupt right turn into a horizontal line. This is what makes me think they converge.

Because each step produces a square-free denominator whose factors are the primes in that span, every step will be co-prime to all other steps and to the denominator of the sum. The numbers can be summed in any order. Because the denominators are made up of the product of the primes for each span, occasionally step n will be less than step n+1. But, the overall trend is decreasing.

I have validated each sum to over 500,000 unchanging digits after about 10-20 steps.

My questions are:

1) What is required to prove convergence?

2) What would it take to prove that a prime-free span cannot occur for the pronic series?

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