91Ó°ÊÓ

The limit of a sequence is a fundamental concept in calculus and real analysis. It refers to the value that the terms of a sequence approach as the sequence progresses to infinity. In simpler terms, as the index number of a sequence grows larger, the sequence will get closer and closer to its limit, if such a limit exists.
For the sequence given by \( a_n = \frac{(10/11)^n}{(9/10)^n + (11/12)^n} \), we aim to find whether it converges. Convergence means the sequence approaches a finite number. By simplifying and analyzing the terms, it is established that \((11/12)^n\) dominates, and thus allows us to simplify the sequence further.
By observing that as \( n \to \infty \), the dominant term \((11/12)^n\) makes \((9/10)^n\) negligible, the sequence simplifies to \(\left(\frac{120}{121}\right)^n\), indicating that the sequence converges to 0 because \(\left(\frac{120}{121}\right) < 1\). Hence, as \( n \to \infty \), \( a_n \to 0 \).

Exponential functions play an important role in many mathematical contexts, including sequence convergence. An exponential function is typically in the form \( b^x \) where \( b \) is a constant base, and \( x \) is the exponent. When used in sequences, exponential terms determine the growth or decay of the sequence.
In this exercise, both the numerator \((10/11)^n\) and the two terms in the denominator \((9/10)^n + (11/12)^n\) are exponential functions. What makes them interesting is their bases:

• \((10/11) < 1\), indicating decay as \( n \) increases.
• \((9/10) < (11/12)\), showing comparative rates of decay with \( (11/12) \) decaying slower.

As \( n \) goes to infinity, terms with bases between 0 and 1 will shrink towards zero. Such behavior is key to simplifying sequences and finding their limits. Observing which term dominates further influences the simplification process.

Determining the dominance of terms within a sequence is crucial for understanding its behavior for large \( n \). When dealing with a sequence that has multiple terms in its expression, especially exponential ones, we look at which term grows or shrinks the slowest.
In the given sequence, comparing \((9/10)^n\) and \((11/12)^n\), we observe that \((11/12)^n\) is the dominant term because it shrinks slower than \((9/10)^n\). Thus, as \( n \) becomes larger, the influence of \((9/10)^n\) diminishes, allowing \((11/12)^n\) to effectively dictate the behavior of the denominator.
By focusing on dominant terms, we can simplify sequences significantly. In our sequence \( a_n \), the dominance of \((11/12)^n\) is why we approximated the denominator by \( (11/12)^n \), leading to an easier computation of \( a_n \) approaching 0 as \( n \to \infty \). This analysis of dominance is a powerful tool for confirming convergence.