Determine whether the sequences
The Sequence an
Using the definition of convergence of an infinite sequence, we would evaluate the following limit:
Because this limit evaluates to a single finite number, the sequence converges.
The Sequence bn
While the sequence an converges to 1/2, bn does not converge because its terms do not approach any number. Instead, the terms in the sequence oscillate between -1 and +1.
The sequence bn does not converge to a unique number, and so bn does not converge.