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

Jump to navigation Jump to search

## Example

Determine whether the sequences

{\displaystyle {\begin{aligned}a_{n}={\frac {1}{2+{\frac {1}{n}}}}\end{aligned}}}

and ${\displaystyle b_{n}=(-1)^{n}}$ converge.

## Complete Solution

### The Sequence an

Using the definition of convergence of an infinite sequence, we would evaluate the following limit:

{\displaystyle {\begin{aligned}\lim _{n\rightarrow \infty }a_{n}=\lim _{n\rightarrow \infty }{\frac {1}{2+{\frac {1}{n}}}}={\frac {1}{2}}\end{aligned}}}

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.