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Understanding the convergence of series is fundamental in the study of infinite series. Simply put, if the sum of an infinite series approaches a specific value as the number of terms increase, we say the series converges. To ascertain convergence, we have various tests, one of which is the Limit Comparison Test. This test is particularly beneficial when dealing with a complex series whose convergence isn't immediately apparent.

The Limit Comparison Test involves comparing the series in question to a simpler 'comparison series' that we already understand well. If the ratio of the nth terms of the two series approaches a positive, finite limit as n goes to infinity, and the comparison series converges, then the original series converges as well.

A p-series is an infinite series of the form \[\sum_{k=1}^{\rm{\infty}} \frac{1}{k^p}\], where p is a constant. The convergence of a p-series is determined by the value of p. If p greater than 1, the series converges; if p is less than or equal to 1, the series diverges. This behavior is due to the integral test, another useful method for testing convergence, which shows us that the integral of 1/x^p converges only when p > 1.

When we encounter a series that is not immediately recognizable as a p-series, we might compare it to a p-series to use the properties of the p-series to determine convergence of the original series.

An infinite series is a sum of an infinite number of terms. They can be daunting, because unlike finite sums, we cannot simply 'add up' all the terms. Instead, we look for patterns and use tests to determine if adding infinitely many terms will 'settle down' to a finite value. The series given in the exercise, \[\sum_{k=1}^{\rm{\infty}} \frac{1}{\sqrt{k^3-k+1}}\], is an example of such a series where direct calculation of the sum isn't feasible. Instead, we use tests like the Limit Comparison Test to determine the behavior of such infinite sums.

The term 'asymptotic behavior' refers to the behavior of a function as the variable goes to infinity. In terms of series, we often look at how the nth term behaves as n becomes very large to understand the properties of the entire series. In the context of the Limit Comparison Test, we look at the leading term because it dominates the behavior of the function for large values of n. In the exercise, the \(k^3\) term in the denominator is the dominant term. When comparing the terms of two series as n approaches infinity, we often focus on these dominant, or 'asymptotic', terms to simplify our comparison and reach a conclusion about convergence.