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Understanding computational complexity is fundamental for analyzing how efficient an algorithm is. It refers to the amount of resources, such as time or space, that an algorithm requires to solve a problem as the size of the input, denoted by 'n', increases. Big O notation, exemplified as \( \mathrm{O}(n^3) \) in the given exercise, is a mathematical notation used to classify algorithms according to their worst-case performance. It provides an upper bound on the growth rate of an algorithm's running time or space requirements in relation to the increase in input size.

As we look at the summation formula in the exercise, \( \sum_{i=1}^{n} i(i+1) = \mathrm{O}\left(n^{3}\right) \), we are essentially analyzing the computational complexity of summing the first 'n' integers multiplied by their successive numbers. The solution process involves breaking down the series and solving it with the well-known summation formulas, leading to a conclusion about the algorithm's performance — in this case, that it doesn't exceed the cubic growth \( n^3 \) as 'n' becomes large.

Summation formulas are indispensable tools when dealing with series in mathematics. They provide the totals of sequences generated by simple patterns, like the running total of the first 'n' natural numbers or the sum of their squares. In the context of the exercise, two specific summation formulas are used: \( \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \) for the sum of squares, and \( \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \) for the sum of the first 'n' natural numbers.

These formulas simplify the process of calculating the sum of complex series by decomposing them into more basic parts, making it easier to obtain a final result without having to perform the sum term by term, which is particularly helpful for large values of 'n'.

Asymptotic analysis is a way of describing the limiting behavior of a function — what happens as the argument of the function grows without bound. In computer science, it is used to describe the performance of algorithms, particularly within the context of big O notation.

Returning to the exercise, after we find the sum of the series using summation formulas, asymptotic analysis is applied. This means comparing the growth rate of our function to \( n^3 \) as 'n' tends to infinity to determine whether the sum is confined within that complexity class. The function \( R(n) \) is defined as a ratio of the sum and \( n^3 \) to investigate its behavior as 'n' becomes very large. This leads to determining that the sum behaves asymptotically as \( n^3 \) because the limit of the ratio does not grow without bounds but settles to a constant.

Limits are a fundamental concept in calculus and are closely related to the idea of asymptotic behavior. They describe the value that a function approaches as the input (or index, in the case of a series) approaches some value. In our step-by-step solution for the exercise, the limit of \( R(n) \) as 'n' goes to infinity is computed to gauge the behavior of the sum relative to \( n^3 \).

The final step involves simplifying this limit, and by doing so, it is observed that it approaches a constant value of \( \frac{1}{3} \) as 'n' grows larger. This is exactly how limits help us in the field of computational complexity: they offer a means to confirm if, and how snugly, the given function fits in a complexity class determined by big O notation. The finding that the limit is a constant and not infinite or unbounded indicates that the sum is indeed \( \mathrm{O}(n^{3}) \) — a crucial insight that can only be obtained through the application of limits.