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!)+...

ritualizi6zk 2022-11-20 Answered
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?
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Answers (1)

Quinten Cervantes
Answered 2022-11-21 Author has 13 answers
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...
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