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Kiana Dodson

Kiana Dodson

Answered question

2022-06-23

n = 1 x sin n is never greater than 2?

Answer & Explanation

Arcatuert3u

Arcatuert3u

Beginner2022-06-24Added 30 answers

Note that
n = 1 N sin ( n ) = Im ( n = 1 N e i n ) = Im ( e i e i ( N + 1 ) 1 e i ) = Im ( e i ( N + 1 ) / 2 sin ( N / 2 ) sin ( 1 / 2 ) ) = sin ( N + 1 2 ) sin ( N / 2 ) sin ( 1 / 2 ) csc ( 1 / 2 ) 2.08582964293349
As pointed out by ClementC., we can write
sin ( N + 1 2 ) sin ( N / 2 ) = 1 2 ( cos ( 1 / 2 ) cos ( N + 1 / 2 ) )
to reveal that
n = 1 N sin ( n ) = cos ( 1 / 2 ) cos ( N + 1 / 2 ) 2 sin ( 1 / 2 ) cos ( 1 / 2 ) + 1 2 sin ( 1 / 2 ) 1.95815868232297
Moreover, we find that
n = 1 N sin ( n ) = cos ( 1 / 2 ) cos ( N + 1 / 2 ) 2 sin ( 1 / 2 ) cos ( 1 / 2 ) 1 2 sin ( 1 / 2 ) 0.127670960610518

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