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To evaluate a sequence effectively, you need to understand its behavior across different ranges of its components, especially as the variables grow larger. In our case, we evaluate the sequence \( a_n = n^2 \cdot (1 - \cos (1/n)) \) for values of \( n \) from 1 to 15. Calculating these provides insight into how \( a_n \) behaves at smaller \( n \). For example, for \( n = 1 \), the calculation gives 0.4597. You continue this for each \( n \) up to 15.

Yet, to understand the sequence more deeply, it's important to evaluate it at much larger scales of \( n \), such as \( n = 10^k \) for \( k = 1,2,3,4 \). This involves recalculating \( a_n \) at these larger values (like \( n = 10, 100, 1000, 10000 \)). Such evaluations reveal how \( a_n \) behaves and helps in hypothesizing its limit, \( \ell \). The behavior gradually settles towards a particular value which indicates the sequence's limit as \( n \) approaches infinity.

The Taylor series is an invaluable tool in mathematics, particularly in approximating functions. By expressing \( \cos(x) \) as a Taylor series, we get: \( \cos(x) \approx 1 - \frac{x^2}{2} + \frac{x^4}{24} \cdots \). This expansion allows us to simplify and understand the limiting behavior of sequences.

In the case of our sequence \( a_n = n^2 \cdot (1 - \cos (1/n)) \), substituting the Taylor expansion, particularly \( \cos(1/n) \approx 1 - \frac{1}{2n^2} \) can significantly simplify calculation as \( n \) becomes very large. As such, \( a_n \approx n^2 \cdot \left(1 - \left(1 - \frac{1}{2n^2}\right)\right) \), simplifying to \( \frac{1}{2} \) for a very large \( n \). This demonstrates that the sequence's limit \( \ell \) is \( \frac{1}{2} \), which aligns with our hypothesis.

Graphical analysis is a powerful method to visually interpret the behavior of functions and sequences. By graphing \( f(x) = \frac{1 - \cos(x)}{x^2} \) for \( 0 < x \leq 1 \), you can observe how the function behaves as \( x \) approaches zero.

The function's value near \( x \rightarrow 0 \) simplifies to \( \frac{1}{2} \), which helps confirm the calculated limit of the sequence \( a_n \). The graph shows a stabilization towards \( \frac{1}{2} \), further solidifying our understanding that as \( n \rightarrow \infty \), \( a_n \) does indeed approach its limit \( \ell \) of \( \frac{1}{2} \).

Such visual aids are critical in comprehending complex mathematical relationships at a glance, enhancing our comprehension of theoretical and practical aspects of sequences.