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hawatajwizp 2022-06-26 Answered
Show that lim n i = n B n arctan ( i ϕ ) arccos ( ϕ i ) = B 2 π
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britspears523jp
Answered 2022-06-27 Author has 28 answers
Denote f ( x ) = log arctan ( x ϕ ) arccos ( ϕ x ) . Expanding into series
f ( x ) = 2 π x + O ( x 2 ) , x ,
and using the Euler's approximation for the harmonic sums
i = 1 n 1 i = γ + log n + O ( 1 / n ) ,
where γ is the Euler's constant, gives
log lim n i = n B n arctan ( i ϕ ) arccos ( ϕ i ) = lim n i = n B n f ( i ) = lim n [ 2 π ( i = 1 B n 1 i i = 1 n 1 1 i ) + O ( n 1 ) ] =
= lim n [ 2 π ( log B n log n ) + O ( n 1 ) ] = 2 π log B .
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