I have to evaluate lim n → ∞ ( − 1 ) n − 1...

Chant6j

Chant6j

Answered

2022-07-02

I have to evaluate
lim n ( 1 ) n 1 sin ( π n 2 + 0.5 n + 1 ) w h e r e n ϵ N

Answer & Explanation

Maggie Bowman

Maggie Bowman

Expert

2022-07-03Added 14 answers

The problem is that as n , n 2 + 0.5 n + 1 diverges, and n 2 + 0.5 n + 1 n 0
. We have
lim n n 2 + 0.5 n + 1 n = lim n ( n + 0.25 ) 2 + 1 0.25 2 n = lim n ( n 2 + 0.5 n + 1 n ) ( n 2 + 0.5 n + 1 + n ) n 2 + 0.5 n + 1 + n = lim n 0.5 n + 1 n 2 + 0.5 n + 1 + n = 1 4
Esmeralda Lane

Esmeralda Lane

Expert

2022-07-04Added 7 answers

Before 1 + 0.5 n + 1 n 2 can "become" 1, it is greater than 1. And so when it multiplies by n 2 , you get something greater than n 2 . And it turns out that it is still greater enough than n 2 to make a difference.
Use n 2 + 0.5 n + 1 = ( n + 0.25 ) 2 + 15 16 . Then you have ( n + 0.25 ) 1 + 15 16 ( n + 0.25 ) 2 . At this point your intuition works out to be correct, and 1 + 15 16 ( n + 0.25 ) 2 goes to 1 quickly enough. But more should be done to prove this.

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