Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.03 Convergence of Infinite Sequences Example

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Determine whether the sequences

and converge.

Complete Solution

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.