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February 25

Do these converge to nice numbers?

Do 1⁄(2*3) + 1⁄(3*5) + 1⁄(5*7) + ...

and 1⁄(2*3) + 1⁄(5*7) + 1⁄(11*13) + ...

(where the numbers in the denominators are primes) converge to known numbers? That is, I have their approximate numerical values - do they converge to something concise? Bubba73 ^{You talkin' to me?} 00:52, 25 February 2018 (UTC)[reply]

Not that I know of, but the 1st one is OEIS sequence A210473, and the 2nd one is related to OEIS sequence A089581. Iffy★Chat -- 10:45, 25 February 2018 (UTC)[reply]

Thanks. Bubba73 ^{You talkin' to me?} 17:41, 25 February 2018 (UTC)[reply]

I don't know whether you tried it, but if you have good numerical values, you can check the Inverse Symbolic Calculator to obtain a guess what the number's exact value could be. —Kusma (t·c) 20:26, 25 February 2018 (UTC)[reply]

The first sum (let's denote it

S

) can be calculated in the following way:

S=\sum \limits _{n=0}^{\infty }{\frac {1}{(2n+1)(2n+3)}}-{\frac {1}{6}}

.

On the other hand:

S'=\sum \limits _{n=-\infty }^{\infty }{\frac {1}{(2n+1)(2n+3)}}=2*\sum \limits _{n=0}^{\infty }{\frac {1}{(2n+1)(2n+3)}}-1

from where it is clear that

S={\frac {S'+1}{2}}-{\frac {1}{6}}

.

The value of

S'

can be calculated using the following integral

S'={\frac {1}{2i}}\oint \limits _{C}{\frac {\cot(\pi z)}{(2z+1)(2z+3)}}dz

,

where the closed contour

C

is chosen to go from

-\infty

to

\infty

below the real axis and back above it. The only poles of the function under the integral are those of cotangent. Therefore

S'=0

. The final result is

S=1/3

.

The second sum can be calculated in a similar way.

Ruslik_Zero 20:39, 26 February 2018 (UTC)[reply]

The denominators in the series are actually the primes. Bubba73 ^{You talkin' to me?} 20:47, 26 February 2018 (UTC)[reply]

... so the first sum is probably closer to 0.3 than to 0.33, but I can't prove it. Dbfirs 08:49, 27 February 2018 (UTC)[reply]

Also, I'm pretty sure you can avoid complex analysis and get S = 1/3 using telescoping sums. In case anyone cares

1⁄(1*3) + 1⁄(5*7) + 1⁄(9*11) + ... = π⁄8. — Preceding unsigned comment added by RDBury (talk • contribs) 10:51, 27 February 2018 (UTC)[reply]

@RDBury: Can you give a reference for that result? I’d like to add it to our article List of formulae involving π. Thanks. Loraof (talk) 17:07, 27 February 2018 (UTC)[reply]

See Leibniz formula for π#Convergence. --RDBury (talk) 10:53, 28 February 2018 (UTC)[reply]

Yes, numerically I got about 0.30109317 for the first one and about 0.21042575 for the second one. Bubba73 ^{You talkin' to me?} 15:40, 27 February 2018 (UTC)[reply]

Sorry, I missed that they are primes. But you should explain better how these sequences are constructed. Do they involve only prime pairs? Ruslik_Zero 18:13, 27 February 2018 (UTC)[reply]

No, consecutive primes. The first one: 1st and 2nd primes, then 2nd and 3rd, then 3rd and 4th, etc. The second one: 1st and 2nd primes, then 3rd and 4th, then 5th and 6th, etc. Bubba73 ^{You talkin' to me?} 18:24, 27 February 2018 (UTC)[reply]