[an error occurred while processing this directive] [an error occurred while processing this directive]

Consider the sequence

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

and

Does the sequence a_{n} have a limit?

Compute a few terms of the sequence to see that a_{1} =
0, a_{2} = -1/2,
a_{3} = -4/3,
a_{4} = 5/4,
a_{5} = 4/5,
and so on.

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

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

The sequence does not have a limit.

Back to main section with problem statement.

[an error occurred while processing this directive]

Last modified: [an error occurred while processing this directive]