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Ayaan Barr

Ayaan Barr

Answered question

2022-07-07

Proving lim n + sin ( 2 π n 2 + n ) = 0

Answer & Explanation

Alec Blake

Alec Blake

Beginner2022-07-08Added 11 answers

Since sin is periodic with period 2π, you know that
( n N ) : sin ( 2 π n 2 + n ) = sin ( 2 π n 2 + n 2 π n ) = sin ( 2 π ( n 2 + n n ) ) .
But
lim n n 2 + n n = 0.
Lucian Maddox

Lucian Maddox

Beginner2022-07-09Added 8 answers

I would write
sin ( 2 π n 2 + n ) = sin ( 2 π n 1 + n n 2 ) = sin ( 2 π n ( 1 + n 2 n 2 + o ( n 3 / 2 ) ) )
= sin ( π n n + o ( n 1 / 2 ) )
The argument of the sinus tends to 0, and therefore, the whole sequence also.

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