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George Bray 2022-06-25 Answered
Find the value of:
k = 1 a r c c o t ( 2 k + 1 + 1 2 k )
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Answers (2)

lorienoldf7
Answered 2022-06-26 Author has 19 answers
You know
cot ( a b ) = cot a . cot b + 1 cot b cot a
so
a r c c o t ( x . y + 1 y x ) = a r c c o t ( y ) a r c c o t ( x )
k = 1 a r c c o t ( 2 k + 1 + 1 2 k ) = k = 1 a r c c o t ( 2 k + 1 .2 k + 1 2 k ) =
you can rewrite 2 k as 2 k + 1 2 k so you will have
k = 1 a r c c o t ( 2 k + 1 .2 k + 1 2 k + 1 2 k ) = k = 1 a r c c o t ( 2 k ) a r c c o t ( 2 k + 1 ) = k = 1 f ( k ) f ( k + 1 ) = a r c c o t ( 2 ) a r c c o t ( ) = a r c c o t ( 2 ) 0
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Sattelhofsk
Answered 2022-06-27 Author has 5 answers
HINT:
See Are a r c c o t ( x ) and arctan ( 1 / x ) the same function?
Can you use
2 k 2 2 k + 1 + 1 = 2 k + 1 2 k 1 + 2 k 2 k + 1
to find
arctan 2 k 2 2 k + 1 + 1 = arctan ( 2 k + 1 ) arctan ( 2 k )
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