91Ó°ÊÓ

The Squeeze Theorem is a fundamental concept used in calculus, especially useful when dealing with sequences and limits. In simple terms, the Squeeze Theorem states that if you have two sequences or functions "squeezing" a third sequence/function between them, and they all converge to the same limit, then the middle one must converge to that limit as well. This technique is particularly helpful when a sequence itself is complicated to directly deal with.
To apply the Squeeze Theorem:

• You first establish two simpler sequences or functions that "sandwich" the given sequence.
• Show that both these sequences converge to the same limit as the bigger sequence.
• Finally, conclude that your given sequence converges to that same limit.

In our specific case with the sequence \( a_n = \frac{\sin n}{n} \), we know \(-1 \leq \sin n \leq 1\). Dividing the whole inequality by \( n \), it becomes \(-\frac{1}{n} \leq \frac{\sin n}{n} \leq \frac{1}{n}\). Both \(-\frac{1}{n}\) and \(\frac{1}{n}\) go to zero as \( n \) approaches infinity, so by the Squeeze Theorem, \( \frac{\sin n}{n} \) also converges to zero.

The concept of the limit of a sequence plays a crucial role in determining whether a sequence converges or diverges. A sequence converges if its terms approach a specific number, called the limit, as you move to infinity. Conversely, if a sequence does not approach any particular number, it diverges.
For sequences, the notation \( \lim_{n \to \infty} a_n = L\) indicates that as \( n \) becomes very large, the terms \( a_n \) come close to \( L\), the limit.
In the sequence \( \frac{\sin n}{n} \), we see that as \( n \) increases:

• The numerator, \( \sin n \), remains bounded between -1 and 1.
• The denominator, \( n \), grows indefinitely.

Dividing by an ever-increasing number tends to "shrink" the fraction towards zero. Hence, the sequence \( \frac{\sin n}{n} \) converges to the limit 0.

Oscillating functions are functions that move between two values repeatedly and do not settle at a particular point as their input changes. The sine function, \( \sin n \), is a classic example of an oscillating function. It varies between -1 and 1 for all integer inputs, creating waves of regular intervals.
When an oscillating function, like \( \sin n \), is part of a sequence such as \( \frac{\sin n}{n} \), its endless ups and downs can complicate convergence analysis at first glance. However, because we know the oscillation is limited to a certain range, it makes it easier to use tools like the Squeeze Theorem to analyze its behavior as part of more complex expressions.
In convergence discussions, understanding that \( \sin n \) itself remains limited helps us reason that multiplying or dividing such behavior by other terms (like \( n \)) can result in a more manageable function or sequence. Hence, even though \( \sin n \) oscillates, \( \frac{\sin n}{n} \) does not oscillate towards infinity. Instead, it converges to a stable limit, which is zero in this case.