Question

In Exercises \(27-30,\) find the limit of \(s(n)\) as \(n \rightarrow \infty\) $$ s(n)=\frac{81}{n^{4}}\left[\frac{n^{2}(n+1)^{2}}{4}\right] $$

Short answer

The limit of the sequence \(s(n)\) as \(n\) approaches infinity is \(\frac{81}{4}\).

Step-by-step solution

Simplify the equation

To start with, the given equation can be simplified. Namely, the term \(\frac{81}{n^{4}}\left[\frac{n^{2}(n+1)^{2}}{4}\right]\) can be rewritten as \(\frac{81n^{2}(n+1)^{2}}{4n^{4}}\)

Further simplification

Further we simplify to \(\frac{81n^{2}(n^{2}+2n+1)}{4n^{4}}\), which becomes \(\frac{81n^{4}+162n^{3}+81n^{2}}{4n^{4}}\)

Evaluate the limit

For each term in the equation we give the limit by dividing each term in the numerator and the denominator by \(n^{4}\), to yield \(\frac{81+\frac{162}{n}+\frac{81}{n^{2}}}{4}\). As \(n\) tends to infinity, any term with \(n\) in the denominator will go towards 0 leading to \(\lim_{{n \rightarrow \infty}} s(n) = \frac{81}{4}\)

Key concepts

Infinite Limits

Understanding infinite limits is vital when studying calculus. An infinite limit refers to the behavior of a sequence or function as its terms grow without bound, either positively or negatively. For instance, when we say the limit of a sequence as n approaches infinity, we are interested in what value the sequence approximates, if any, as the terms become arbitrarily large.

When determining an infinite limit for sequences, if the terms increase or decrease without approaching a particular value, we say the limit is infinity or negative infinity respectively. However, not all sequences have infinite limits; some may converge to a specific number. In our exercise, we observe that as n grows without bounds, the terms that include n in the denominator approach zero, therefore leading to a finite limit rather than an infinite one.

Sequence Convergence

The concept of sequence convergence is a cornerstone of calculus. A sequence converges if it approaches a fixed value, known as the limit, as its terms go on indefinitely. To determine if a sequence converges, we analyze its behavior as n tends to infinity. If the sequence's terms get arbitrarily close to a specific number, we say the sequence converges to that number.

In the context of our exercise, we are calculating the limit of the sequence s(n) as n tends towards infinity. Through the process of simplification and elimination, it becomes clear that the sequence does indeed converge. The polynomial terms with n in the numerator and denominator cancel out or diminish to zero, leaving us with a constant value, signifying the convergence of the sequence to that constant.

Polynomial Simplification

The art of polynomial simplification involves reducing complex polynomial expressions to simpler forms, which can often reveal properties about the expression that were not initially apparent. This is done by combining like terms, factoring, and canceling out terms where possible.

In our step-by-step solution, we demonstrated polynomial simplification by expanding the expression to comprehend its behavior better as n approaches infinity. The process included canceling out powers of n to simplify the fraction, which was pivotal in determining the limit of the sequence. Without such simplification, it would be more challenging to analyze the behavior of the sequence and to conclude definitively whether it converges or diverges.

Exercise Improvement Advice

It is important for students to clearly understand each step taken during simplification. Showing intermediate steps could enhance comprehension, allowing students to comfortably follow along with the algebraic manipulations and grasp the underlying mathematical principles.