- #1
Rudy Toody
- 22
- 0
I have found an interesting infinite sum that appears to converge to a number.
In fact, I have created four of these using slightly different rules.
Since, all of the odd primes are involved in each series, I know that the series are infinite. I will confirm this by using Euler's Prime Product in place of the sum at each step. I should get Pi^2 / 6.
Now, I have one more series that is nearly identical to one of the above, but it drops an occasional prime (about one prime for every 8 steps.) The two series are in lock-step for 7 out of 8 steps.
That series also seems to converge to a number.
Question1: If I can prove convergence of the series that contains all of the primes, and if I can prove that when using the ratio test that they converge, would that indicate that the second series is also infinite? I know it would mean that it is convergent.
Question2: Can I use the Prime Product as one (or both) elements of the ratio test, or does that only work for sums?
In fact, I have created four of these using slightly different rules.
Since, all of the odd primes are involved in each series, I know that the series are infinite. I will confirm this by using Euler's Prime Product in place of the sum at each step. I should get Pi^2 / 6.
Now, I have one more series that is nearly identical to one of the above, but it drops an occasional prime (about one prime for every 8 steps.) The two series are in lock-step for 7 out of 8 steps.
That series also seems to converge to a number.
Question1: If I can prove convergence of the series that contains all of the primes, and if I can prove that when using the ratio test that they converge, would that indicate that the second series is also infinite? I know it would mean that it is convergent.
Question2: Can I use the Prime Product as one (or both) elements of the ratio test, or does that only work for sums?