Yahoo Answers: Answers and Comments for Lim (n to inf) [ln (1+3n+4n^2)ln(5+6n+2n^2)]? [Mathematics]
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From Ava
enCA
Sun, 18 Aug 2019 19:41:26 +0000
3
Yahoo Answers: Answers and Comments for Lim (n to inf) [ln (1+3n+4n^2)ln(5+6n+2n^2)]? [Mathematics]
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From Captain Matticus, LandPiratesInc: I'd combine them and simplify
ln((4n^2 + ...
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Sun, 18 Aug 2019 19:55:14 +0000
I'd combine them and simplify
ln((4n^2 + 3n + 1) / (2n^2 + 6n + 5)) =>
ln(n^2 * (4 + 3/n + 1/n^2) / (n^2 * (2 + 6/n + 5/n^2))) =>
ln((4 + 3/n + 1/n^2) / (2 + 6/n + 5/n^2))
n goes to infinity
ln((4 + 0 + 0) / (2 + 0 + 0)) =>
ln(4/2) =>
ln(2)
a(n) = ln(1 + 3n + 4n^2)  ln(5 + 6n + 2n^2)
n goes to infinity
a(n) = ln(inf)  ln(inf) = inf  inf
That's indeterminate, so we can use L'Hopital's rule
a(n) = ln((4n^2 + 3n + 1) / (2n^2 + 6n + 5))
e^(a(n)) = (4n^2 + 3n + 1) / (2n^2 + 6n + 5)
f(n) = 4n^2 + 3n + 1
f'(n) = 8n + 6
g(n) = 2n^2 + 6n + 5
g'(n) = 4n + 6
f'(n) / g'(n) =>
(8n + 6) / (4n + 6) =>
2 * (4n + 3) / (2 * (2n + 3)) =>
(4n + 3) / (2n + 3)
We get inf/inf, so we can use L'hopital's rule again
f(n) = 4n + 3
f'(n) = 4
g(n) = 2n + 3
g'(n) = 2
f'(n) / g'(n) =>
4/2 =>
2
e^(a(n)) = 2
a(n) = ln(2)
So what went wrong is that you did your math incorrectly. I just used L'hopital's rule and got ln(2)

From JOHN: Answer
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Sun, 18 Aug 2019 20:25:51 +0000
Answer

From ted s: you likely made an error since L'H would y...
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Sun, 18 Aug 2019 19:46:51 +0000
you likely made an error since L'H would yield { on the rational expression } 8 / 4 ....= 2