Evaluate <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-ORD"> n

Jase Howe

Jase Howe

Answered question

2022-06-08

Evaluate n = 1 cot 1 ( 2 n 2 )

Answer & Explanation

America Barrera

America Barrera

Beginner2022-06-09Added 23 answers

I wrote the expression as
∑n=1∞tan−1(12n2)
next step is
n = 1 tan 1 ( 1 2 n 2 ) = n = 1 tan 1 ( 2 n + 1 ( 2 n 1 ) 1 + ( 2 n + 1 ) ( 2 n 1 ) )
n = 1 tan 1 ( 2 n + 1 ( 2 n 1 ) 1 + ( 2 n + 1 ) ( 2 n 1 ) ) = n = 1 tan 1 ( 2 n + 1 ) tan 1 ( 2 n 1 ) = π 4
Extrakt04

Extrakt04

Beginner2022-06-10Added 5 answers

With
2 n 2 = ( 1 n 1 + 1 ) ( 1 n + 1 ) + 1 ( 1 n 1 + 1 ) ( 1 n + 1 )
we obtain
cot 1 ( 2 n 2 ) = cot 1 ( 1 n + 1 ) cot 1 ( 1 n 1 + 1 )
Hence
1 N cot 1 ( 2 n 2 ) = cot 1 ( 1 N + 1 ) .

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