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., \[ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64},\dots\]
Geometric sequence 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 \[ \lim_{n\to\infty} z^n = 0 \]
