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

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### The Sequence

### The Sequence

## Example

Determine whether the sequences

and converge.

## Complete Solution

### The Sequence *a*_{n}

_{n}

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 *b*_{n}

_{n}

While the sequence *a _{n}* converges to 1/2,

*b*does not converge because its terms do not approach any number. Instead, the terms in the sequence oscillate between -1 and +1.

_{n}The sequence *b _{n}* does not converge to a unique number, and so

*b*does not converge.

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