Is the sequence defined by $a_n = 1 / 2^n$ convergent or divergent?
The given sequence is defined by the general term \(a_n = \frac{1}{2^n}\) for \(n = 1, 2, 3, \dots\). We want to determine if this sequence converges or diverges. A sequence \(\{a_n\}\) is defined as convergent if the limit \(\lim_{n \to \infty} a_n\) exists and is a finite number \(L\). If the limit does not exist or is infinite, the sequence is said to be divergent.
The general term can be rewritten as \(a_n = \left(\frac{1}{2}\right)^n\). This is a geometric sequence of the form \(r^n\) where the common ratio is \(r = \frac{1}{2}\). A fundamental property of geometric sequences states that the sequence \(r^n\) converges to \(0\) if \(|r| < 1\), and it diverges if \(|r| \ge 1\) (with the exception of \(r=1\) where it converges to 1). In this case, since \(r = \frac{1}{2}\), we have \(|r| = \frac{1}{2} < 1\), which satisfies the condition for convergence.
To confirm this, we evaluate the limit as \(n\) approaches infinity:
\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(\frac{1}{2}\right)^n = 0 \]
Since the limit is a finite number (0), the sequence \(\{a_n\}\) is convergent. Therefore, the correct option is (a).
A common real-life example of a convergent sequence like this is the concept of radioactive decay. If you start with a certain amount of a radioactive substance, it decays at a constant rate over time, with the amount remaining following a geometric pattern. As time passes, the quantity of the substance approaches zero, representing a convergent sequence in a physical context. Similarly, Zeno's paradox, where you repeatedly cover half the remaining distance to a goal, models this mathematical behavior.