I need to evaluate a series of a function that switches sign in the following way: <mtable colum

Monserrat Sawyer 2022-05-24 Answered
I need to evaluate a series of a function that switches sign in the following way:
k = + sgn ( n k ) ( ( 2 n + 1 ) + B sgn ( n k ) ) ( 2 k + 1 )
where B R and n Z
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Answers (1)

Harper Heath
Answered 2022-05-25 Author has 9 answers
I would first shift the sum to j = n k. Then your sum is
k = + sgn ( n k ) ( ( 2 n + 1 ) + B sgn ( n k ) ) ( 2 k + 1 ) = j = + sgn ( j ) 2 j + B sgn ( j ) = j = 1 + 1 2 j + B j = 1 1 2 j B = j = 1 + 1 j + B / 2 = 2 B ψ ( 0 ) ( B / 2 ) ,
where I replaced j j in the second sum in the second line to move to the third line.
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