Does the sequence converge or diverge?
SOLUTION: ; hence the sequence converges to .
BC ONLYExample 3 __
Does the sequence converge or diverge?
SOLUTION: ; hence the sequence converges to 1. Note that the
terms in the sequence , . . . are alternately smaller and larger than 1.
We say this sequence converges to 1 by oscillation.
Example 4 __
Does the sequence converge or diverge?
SOLUTION: Since , the sequence diverges (to infinity).
Example 5 __
Does the sequence an = sin n converge or diverge?
SOLUTION: Because does not exist, the sequence diverges. However,
note that it does not diverge to infinity.
Example 6 __
Does the sequence an = (−1)n + 1 converge or diverge?
SOLUTION: Because does not exist, the sequence diverges. Note that
the sequence 1, −1, 1, −1, . . . diverges because it oscillates.