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

Example

Determine whether the sequences

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

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

Complete Solution

The Sequence a_n

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

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

Because this limit evaluates to a single finite number, the sequence converges.

The Sequence b_n

While the sequence a_n converges to 1/2, b_n does not converge because its terms do not approach any number. Instead, the terms in the sequence oscillate between -1 and +1.

The sequence b_n does not converge to a unique number, and so b_n does not converge.