I need to find the limit point(s) of the sequence whose general term is given by: a n

Manteo2h 2022-06-27 Answered
I need to find the limit point(s) of the sequence whose general term is given by:
a n = 1 1 n + 1 2 ( n 1 ) + + 1 n 1
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Answers (1)

Aiden Norman
Answered 2022-06-28 Author has 16 answers
The sequence, a n , can be written
a n = k = 1 n 1 k ( n k + 1 ) = 1 n + 1 k = 1 n ( 1 k 1 k ( n + 1 ) ) = 2 n + 1 k = 1 n 1 k
It is easy to show that a n is monotonically decreasing and bounded below by 0. So a n converges. Moreover, from the integral test, we have
k = 1 n 1 k log ( n ) + 1
Therefore, we see that
lim n k = 1 n 1 k ( n k + 1 ) = 0

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