25 Mar '08 17:36

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"π 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?

Thanxπ

(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"π 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?

Thanxπ