Question about pi (I don't know the answer)

See the following interesting 3 series:

(1*2)/(1*3) + (1*2*3)/(1*3*5) + (1*2*3*4)/(1*3*5*7) + (1*2*3*4*5)/(1*3*5*7*9) + ... = pi/2

(1*2)/(3*5) + (1*2*3)/(3*5*7) + (1*2*3*4)/(3*5*7*9) + (1*2*3*4*5)/(3*5*7*9*11) + ... = pi/2 - 4/3

(1*2)/(5*7) + (1*2*3)/(5*7*9) + (1*2*3*4)/(5*7*9*11) + (1*2*3*4*5)/(5*7*9*11*13) + ... = -3pi/2 + 24/5

Which can be written also like this:

(1)/(1) + (1*2)/(1*3) + (1*2*3)/(1*3*5) + (1*2*3*4)/(1*3*5*7) + (1*2*3*4*5)/(1*3*5*7*9) + ... = pi/2 + 1

(1)/(3) + (1*2)/(3*5) + (1*2*3)/(3*5*7) + (1*2*3*4)/(3*5*7*9) + (1*2*3*4*5)/(3*5*7*9*11) + ... = pi/2 - 1

(1)/(5) + (1*2)/(5*7) + (1*2*3)/(5*7*9) + (1*2*3*4)/(5*7*9*11) + (1*2*3*4*5)/(5*7*9*11*13) + ... = -3pi/2 + 5

Another way to write the third one:

(1)/(3) + (1*1)/(3*5) + (1*1*2)/(3*5*7) + (1*1*2*3)/(3*5*7*9) + (1*1*2*3*4)/(3*5*7*9*11) + ... = -pi/2 + 2

My question is, can this "sequence of series"🙄 be continued, and the sum is always of the form a*pi+b with a,b rational?

(1*2)/(7*9) + (1*2*3)/(7*9*11) + (1*2*3*4)/(7*9*11*13) + (1*2*3*4*5)/(7*9*11*13*15) + ... = ?

(1*2)/(9*11) + (1*2*3)/(9*11*13) + (1*2*3*4)/(9*11*13*15) + (1*2*3*4*5)/(9*11*13*15*17) + ... = ?

Maybe someone has Maple/Mathematica or a similar software which can evalute these series?

Thanx😀

i wish i was smart enough to know what is interesting about this

Originally posted by coquette
i wish i was smart enough to know what is interesting about this

😞

Like a teacher of mine said: "The measure of interest in one thing is how much you try to understand it"

In this case if the result holds it is a very interesting and general result. Of course it won't give us better toast in the morning but if you solve it you'll have the satisfaction of solving a mathematical theorem. 🙂

this is basically sigma( n! / [(2n)!/2^n*n!]) or rather: sigma (2^n*n!*n!/[2n]!) as n goes from 1 to infinity, and then afterwards incrementing the denominator to get later series yes?

i.e. more generally, looking at sigma (2^[n+k]*[n+k]!*n!/[2(n+k)]!) and looking for a convergence to some a*pi +b ?

i haven't spent any time with pen and paper to see if this relates well to an earlier series you know with factoring, but i think the (2n+2k)! in the denominator grows faster than the numerator does, so we should be guaranteed convergence... can't be sure about a*pi +b with a,b rational though

I've managed to prove (in a fairly hand-wavy way) that these series converge to elements in Q[Pi] (the algebraic extension of the rationals by Pi). That doesn't get me any closer to whether that relation is linear, however... it seems plausible, but I haven't been able to prove that yet. Maybe when I get home I'll be able to use Mathematica to get something more for you. Thanks for the interesting problem!

Originally posted by coquette
i wish i was smart enough to know what is interesting about this

We wish you were too!