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These are replies submitted by RLessard

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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