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The Limit Comparison Test is a powerful tool to determine the convergence or divergence of infinite series by comparing them to another series whose behavior is known. If you have two positive series, \( \sum a_n \) and \( \sum b_n \), and the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \) where \( 0 < c < \infty \), then both series either converge or diverge together.
This test is particularly useful when the series involve complicated expressions, as it helps simplify the process by comparing it to a series you already know about.
In the exercise, knowing that \( \lim_{n \to \infty} \left(\frac{a_n}{b_n}\right) = \infty \) implies that \( b_n \) must approach zero faster compared to \( a_n \). This feature is not a typical scenario directly addressed by the Limit Comparison Test, but it helps us understand the relative sizes of the sequences involved.

A convergent series is one where the sum of its terms approaches a finite limit as you sum up infinitely many terms. Mathematically, a series \( \sum a_n \) converges if its sequence of partial sums \( S_N = a_1 + a_2 + \cdots + a_N \) approaches a specific number as \( N \to \infty \).
For a series to converge, the terms \( a_n \) must tend to zero as \( n \to \infty \). However, this condition is necessary but not sufficient alone; the terms must decrease rapidly enough. For instance, the terms in the convergent series \( \sum \frac{1}{n^2} \) decrease faster than those in the divergent harmonic series \( \sum \frac{1}{n} \).
In the given problem, since \( \sum a_n \) converges, it implies that the terms \( a_n \) decrease to zero sufficiently fast. This characteristic assists in understanding the behavior of another series \( \sum b_n \) with respect to it.

A divergent series is one where the partial sum does not approach a finite limit as the number of terms grows. Essentially, the series could grow indefinitely or oscillate without settling on a particular value.
The simplest example is the harmonic series \( \sum \frac{1}{n} \), which diverges because its terms do not decrease rapidly enough to result in a finite sum.
In the exercise's context, \( \lim_{n \to \infty} \frac{a_n}{b_n} = \infty \) suggests, counterintuitively at first glance, that \( \sum b_n \) is not divergent. Instead, since it approaches zero faster than \( a_n \), which forms a convergent series, \( \sum b_n \) is also convergent in this scenario.

The Ratio Test is a method used to assess the convergence of an infinite series \( \sum a_n \). For the test, consider the limit \( L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| \).

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \) or is infinite, the series diverges.
• If \( L = 1 \), the test is inconclusive.

The Ratio Test is particularly useful for series involving factorials or exponential terms because these factors can often be simplified significantly in the ratio \( \frac{a_{n+1}}{a_n} \).
Although not directly used in this exercise, understanding the Ratio Test enriches your toolkit for tackling series questions, complementing techniques like the Limit Comparison Test used in this solution.