91Ó°ÊÓ

A sequence is considered to be bounded if there exists a real number that serves as a limit beyond which the values of the sequence do not go. More formally, a sequence \(a_n\) is bounded if there exist two real numbers \(M\) and \(N\) such that \(N \leq a_n \leq M\) for all \(n\). For complex sequences, a sequence \(((z_n))\) is bounded if the absolute values of the terms \( |z_n| \) are bounded.

In the case of the sequence \( z_n = \frac{(3+4i)^n}{n!} \), the term \( \left|\frac{(3+4i)^n}{n!}\right| = \frac{5^n}{n!} \) as shown in the solution, becomes smaller as \(n\) grows larger. Since this value gets indefinitely close to zero and never exceeds some bound, we assert that \(z_n\) is bounded by any number greater than zero.

A sequence \(((a_n))\) is said to be convergent if it approaches a specific value, called the limit, as \(n\) goes to infinity. Mathematically, a sequence \(a_n\) is convergent if there exists a number \(L\) such that for every \(\epsilon > 0\), there exists an integer \(N\) such that \(|a_n - L| < \epsilon\) for all \(n > N\).

In the exercise sequence \( z_n = \frac{(3+4i)^n}{n!} \), the limit was found by comparing the growth rates of \((3+4i)^n\) and \(n!\). While \( (3+4i)^n \) grows exponentially, \( n! \) grows faster. This led to the conclusion that \( \lim_{n \to \infty} \frac{5^n}{n!} = 0 \), implying \(z_n\) converges to 0.

Limit points, also known as accumulation points, of a sequence are values that the terms of the sequence get arbitrarily close to as the sequence progresses. A sequence can have multiple limit points, or just one, or even none, depending on its properties.

For the sequence given by \( z_n = \frac{(3+4i)^n}{n!} \), we established that the entire sequence converges to 0. This indicates that 0 is the only limit point of this sequence. To say a sequence has a limit point of 0 means that for any \( \epsilon > 0 \), there are infinitely many terms in the sequence within \( \epsilon \) distance of 0.

• For convergent sequences like ours, the limit (0 in this case) is often the sole limit point.
• Non-convergent sequences might have a set of limit points instead.