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When studying sequences in mathematics, one key concept that emerges is the 'limit of a sequence'. This refers to the value that the sequence's terms get closer to as the term number increases indefinitely. For example, consider the sequence defined by the formula \( a_n = \frac{n}{\sqrt[n]{n!}} \). To understand if this sequence converges, we look for the limit as \( n \) approaches infinity.

Convergence means that there is a specific number that the terms of the sequence get infinitely close to. If such a number exists, the sequence is said to converge to that number, also known as the 'limit' of the sequence. The graphical plot can offer insights into the behavior of the sequence and whether terms are 'squeezing' together towards a single value or spreading apart.

For sequence (a), we deduce convergence by showing that as \( n \) increases, the terms of the sequence approach a particular value. This alignment with a definite value is pivotal in understanding the fundamental nature of sequences in calculus.

In higher mathematics, 'Stirling's approximation' is an essential tool for estimating the factorial of large numbers. It is especially useful when factorials are part of complex functions or sequences. The formula states that \( n! \) can be approximated by \( \sqrt{2 \pi n} * \left(\frac{n}{e}\right)^n \), where \( e \) is the base of the natural logarithm.

This approximation becomes incredibly accurate as \( n \) grows large. In our sequence (a), Stirling's approximation is utilized to find the limit of \( \frac{n}{\sqrt[n]{n!}} \) as \( n \) approaches infinity. Through simplifying the expression using Stirling's approximation, we can then calculate the limit, which in this case, yields the value \( e \). This technique is a brilliant example of how a complex concept can be made simpler through a powerful approximation and depicts the synergy between different areas of mathematics.

The 'squeeze theorem' (also known as the 'sandwich theorem') is an invaluable concept in calculus for determining the limits of sequences, especially when the sequence itself is difficult to evaluate directly. The theorem states that if a sequence is 'squeezed' between two other sequences whose limits are the same, then it must also converge to the same limit.

In relation to sequence (b) from the exercise, \( b_n = n^2 \sin(\pi / n) \), we apply the squeeze theorem by showing that \( 0 \leq \sin(\pi / n) \leq 1 \) and consequently \( 0 \leq n^2 \sin(\pi / n) \leq n^2 \). As \( n \) tends to infinity, the term \( n^2 \) increases without bound. Therefore, the limit of the sequence is infinity, confirming divergence.

This theorem is an incredible demonstration of how boundaries can be used to establish limits and facilitates the understanding of sequences' behavior at infinity.