I want to determine the sum of the series π 2 4 2 2 ! − π 4 4 4 4 ! + π 6 4 6 6 ! − π 8 4 8 8 ! + ⋯What I am trying to do is to consider that cos ⁡ ( x ) = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + . . .then − cos ⁡ ( x ) + 1 = x 2 2 ! − x 4 4 ! + x 6 6 ! − . . .Up to this point the expression resembles the one I am looking for, however I have not been able to find the final result, any help?

Question

I want to determine the sum of the series            π      2                      4        2            2      !        −            π      4                      4        4            4      !        +            π      6                      4        6            6      !        −            π      8                      4        8            8      !        +  ⋯What I am trying to do is to consider that  cos  ⁡  (  x  )  =  1  −            x      2              2      !        +            x      4              4      !        −            x      6              6      !        +  .  .  .then  −  cos  ⁡  (  x  )  +  1  =            x      2              2      !        −            x      4              4      !        +            x      6              6      !        −  .  .  .Up to this point the expression resembles the one I am looking for, however I have not been able to find the final result, any help?

Quinten Cervantes · Accepted Answer

It sounds like you are adding      ∑          n      =      1        ∞    (  −  1      )          2      n      +      1                  (      π              /            4              )                  2          n                            (      2      n      )      !      which will fit into that cosine flavor when you set   x  =  π      /    4...

Determine the sum of the series (pi^2)/(4^2 2!)−(pi^4)/(4^4 4!)+(pi^6)/(4^6 6!)−(pi^8)/(4^8 8!)+...

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