Geometric sequences

Limits part 1: Infinite sequences: Standard limits

Geometric sequences

In this section we study sequences such as $b_n = \left( \frac{1}{2} \right)^n$ , i.e.,

$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64},\dots$

A geometric sequence is a sequence for which there exists a nonzero constant $r$ such that $a_{n+1} = r\dot a_n$ for all $n$ . The constant $r$ is called the common ratio of the geometric sequence.

Our sequence is a geometric sequence with common ratio $\tfrac{1}{2}$ because the value of the each element is precisely one half of the value of the previous element: $b_{n} = \tfrac{1}{2}b_{n-1}$ . In forward direction we have: $b_{n+1} = \tfrac{1}{2}\cdot b_n$ . In this example we can squeeze $b_n$ between $a_n = 0$ and $c_n = \frac{1}{n}$ . So the limit of this sequence equals $0$ . In this interactive plot you can examine the sequence $z^n$ for various values of $z$ .

You can discover the following things:

For $z\leq -1$ we get an alternating sequence that does not converge;
For $-1 < z < 1$ the sequence converges to $0$ ;
For $z = 1$ we obtain the constant sequence $1,1,1,1,\dots$ , so then the limit equals $1$ ;
For $z > 1$ the sequence diverges to $\infty$ .

We have the following standard limit:

If $|z|<1$ then

$\lim_{n\to\infty} z^n = 0$