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Calculus Level 3

True or false :

\quad lim ⁡ n → ∞ sin ⁡ ( n 4 ) \displaystyle \lim_{n\to\infty} \sin(n^4) diverges.

True False

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

展豪 張
Apr 29, 2016

Consider the series a n = sin ⁡ ( b n 4 ) a_n=\sin(b_n^4)
Pick b n = 2 n π 4 b_n=\sqrt[4]{2n\pi}
Then a n = 0 ∀ n a_n=0\forall n .
Pick b n = ( 2 n + 1 2 ) π 4 b_n=\sqrt[4]{(2n+\dfrac 12)\pi}
Then a n = 1 ∀ n a_n=1\forall n .
The two 'limit's do not match, so it diverges.

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Corrrectttttt!

Pi Han Goh - 5 years, 1 month ago

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