lim n → ∞ n 2 ( ln ⁡ ( n + 1 ) −...

letumsnemesislh

letumsnemesislh

Answered

2022-07-06

lim n n 2 ( ln ( n + 1 ) ln n ) = 0
But I haven't any ideas how to do it... My calculations shows that this sequence is monotonously decreasing.
I've proved that using inequality ln ( 1 + x a ) a x and double-sided theorem.

Answer & Explanation

furniranizq

furniranizq

Expert

2022-07-07Added 20 answers

n 2 ( ln ( n + 1 ) ln n ) = 2 n ln ( n + 1 ) ln n ln ( n + 1 ) + ln n = 2 n ln ( 1 + 1 n ) ln ( n + 1 ) + ln n 2 n 1 n 2 ln n = 1 2 ln n 0
letumsnemesislh

letumsnemesislh

Expert

2022-07-08Added 6 answers

Since a n := n 2 ( ln ( n + 1 ) ln n ) = 2 n ln ( n + 1 ) ln n ln ( n + 1 ) + ln n = 2 n ln ( 1 + 1 n ) ln ( n + 1 ) + ln n ,
we have
a n = 2 n ( 1 n 1 n 2 + 1 n 3 ) ln ( n + 1 ) + ln n = 2 1 1 n + 1 n 2 ln ( n + 1 ) + ln n
it follows that
lim n a n = 0.

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