Evaluation of ∑ n = 1 ∞ arctan ⁡ ( 4 n n 4 −...

Holetaug

Holetaug

Answered

2022-06-29

Evaluation of n = 1 arctan ( 4 n n 4 2 n 2 + 2 )

Answer & Explanation

Monserrat Cole

Monserrat Cole

Expert

2022-06-30Added 12 answers

Hint. One may write, for n 2
4 n n 4 2 n 2 + 2 = 4 n ( n 2 1 ) 2 + 1 = 4 n ( n 2 1 ) 2 1 + 1 ( n 2 1 ) 2 = 1 ( n 1 ) 2 1 ( n + 1 ) 2 1 + 1 ( n 1 ) 2 ( n + 1 ) 2
giving here
arctan 4 n n 4 2 n 2 + 2 = arctan 1 ( n 1 ) 2 arctan 1 ( n + 1 ) 2 .

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