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These are replies submitted by RLessard

@Mariusz Iwaniuk Thank you ! But I know your solutions.    This is how I found the solution.

Rather, I'm looking for a particular command in Maple (in sumtools or Sumtools) which would give me the answer.

Thanks anyway !

 Thanks !

  @Preben Alsholm   Thanks !    You provided me with 2 solutions .

    The  arrow  and  "option operator,arrow"  are  fine for me .

     Best Regards !

Hi !

Here is the screenshot of my computer with Windows XP and Maple V Release V .

I have also an old computer with Windows 2000  .  Same screenshot ! 

  Best regards !

Sorry for the screenshots !

Mr. Carl  Love , thank you very much !
The trick is to eliminate the term (2*i + 2*k)!  keeping only (2*i)!  and use convert (exp, parfrac).

As you probably guessed, I want to calculate the infinite summation according to the variable k.
The commands in the file "solution.mw"   work fine for different values of k  .
Best Regards 
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