91Ó°ÊÓ

A sequence is considered "bounded" if its terms do not stray too far from a certain range of values. Specifically, for a sequence \( z_n \), if there exists a real number \( M \) such that all terms \( |z_n| < M \), then we say the sequence is bounded. In simple terms, a bounded sequence will not go off towards infinity or negative infinity.

In the given exercise, the sequence is \( z_n = \frac{(-1)^n}{n+i} \). To determine boundedness, we calculate the magnitude of each term, denoted by \( |z_n| \). The calculation is as follows:

• \( |z_n| = \left|\frac{(-1)^n}{n+i}\right| = \frac{1}{\sqrt{n^2+1}} \)

This result indicates the sequence is always between 0 and 1 for all natural numbers \( n \). Thus, it is clear that the sequence does not exceed these bounds, making it a bounded sequence.

Recognizing boundedness is crucial because it assures us that the sequence's values remain manageable and do not diverge."

A sequence is called "convergent" if it approaches a particular number, known as the limit, as the index \( n \) becomes very large, essentially going towards infinity. In formal terms, a sequence \( z_n \) converges to a limit \( L \) if the values of \( z_n \) get arbitrarily close to \(L \) for sufficiently large \( n \).

For the sequence in our exercise, \( z_n = \frac{(-1)^n}{n+i} \), convergence can be analyzed as follows:

• As \( n \) grows, the imaginary component \( i \) becomes negligible compared to \( n \), simplifying the denominator to \( \sqrt{n^2+1} \).
• The term \( \frac{1}{\sqrt{n^2+1}} \) shrinks towards zero.

Therefore, the elements of the sequence \( z_n \) get closer and closer to zero, indicating that this sequence converges.

The significance of identifying a convergent sequence lies in knowing that it reaches a stable value, facilitating predictions of future behavior for large terms."

The concept of "limit points" for a sequence relates to the idea of accumulation values. A limit point is where the terms of a sequence cluster as \( n \) tends to infinity.

For the sequence in question, \( z_n = \frac{(-1)^n}{n+i} \), the investigation into limit points is already simplified by the finding in Step 2: that it converges to zero. Therefore, zero becomes the limit point of the sequence because the values \( z_n \) get infinitesimally close to zero as \( n \) increases.

• The trend is evident in larger terms, as their proximity to zero demonstrates clustering at this point.

Understanding limit points reveals where most terms lie or the sequence "accumulates" to a specific value, offering insights into the sequence's long-term behavior."