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Problem Statement

Consider the sequence

 |_ n/2 _| (- 1)n an = (-1) + --n---

for n = 1, 2, 3.... Here  |_ x _| is the “floor function”, the greatest integer less than or equal to x, so  |_ 1 _| = 1,  |_ 3/2 _| = 1,  |_ 8/3 _| = 2,  |_ -3/2 _| = -2, etc. Find

lim sup an n--> oo

and

linm-->oin o f an.

Does the sequence an have a limit?

Solution

Compute a few terms of the sequence to see that a1 = 0, a2 = -1/2, a3 = -4/3, a4 = 5/4, a5 = 4/5, and so on.

lim supan = 1 n--> oo

and a subsequence converging to 1 is for n = 4j, j = 1, 2, 3,..., where a4j = 1 + 1/(4j).

limn-->i oo nfan = - 1

and a subsequence converging to -1 is for n = 4j - 1, j = 1, 2, 3,..., where a4j-1 = -1 - 1/(4j - 1).

The sequence does not have a limit.


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