Question

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

Short answer

The limit of the sequence \(s(n)\) as \(n \rightarrow \infty\) is \(0.5\).

Step-by-step solution

Simplification

Start by simplifying the sequence. Factor out \(n^2\) in \(s(n)=\frac{1}{n^{2}}\left[\frac{n(n+1)}{2}\right]\), and you get \(s(n) = \frac{n + 1}{2n}\).

Applying the limit

Apply the limit by substituting \(n \rightarrow \infty\) into the simplified sequence, i.e., find \(\lim_{{n \rightarrow \infty}} \frac{n+1}{2n}\).

Solving the limit

Proceed by determining the dominant term in the numerator and the denominator. In this case, both the numerator and the denominator are dominated by the term 'n'. Therefore, divide every term in the numerator and the denominator with 'n', you will get \(\lim_{{n \rightarrow \infty}} \frac{1 + \frac{1}{n}}{2}\).

Further Simplification

As \(n \rightarrow \infty\), the fraction \(\frac{1}{n}\) approaches 0 and we can simplify to get \(\lim_{{n \rightarrow \infty}} \frac{1 + \frac{1}{n}}{2} = \frac{1+0}{2}\).

Key concepts

Sequence Convergence

Understanding the convergence of sequences is crucial in calculus. A sequence converges if, as it extends to infinity, it approaches a specific value, called the limit. Consider the sequence given by the function
\( s(n)=\frac{1}{n^{2}}\left[\frac{n(n+1)}{2}\right] \).
To determine if this sequence converges, we investigate what happens as \( n \) grows without bound, that is, as \( n \rightarrow \infty \).
In the given solution, simplification played a key role in revealing the sequence's behavior at infinity. By factoring out \( n^2 \) and observing how the terms cancel, we obtained a simpler expression \( \frac{n + 1}{2n} \). By applying the limit and considering the behavior of each term separately, we can see that as \( n \) becomes very large, the terms \( \frac{1}{n} \) become insignificant. The sequence's behavior is essentially dictated by the term 'n', showing that \( \lim_{{n \rightarrow \infty}} \frac{n+1}{2n} \) is a constant value.
This step-by-step analysis helps us conclude that the original sequence converges to a specific limit, which is a fundamental aspect of understanding sequence behavior.

Infinite Limits

While many sequences approach a finite limit, some grow without bound as \( n \rightarrow \infty \), called infinite limits. However, in the context of the exercise, we're dealing with a sequence that doesn't trend towards infinity but rather settles on a finite number.
By examining the simplified sequence \( s(n) = \frac{n + 1}{2n} \) and observing its behavior as \( n \) increases, we intuitively predict that the sequence will not approach infinity since the numerator does not grow significantly faster than the denominator.
The concept of infinite limits becomes essential when sequences or functions grow without restraint. In such cases, we might say that the limit of the sequence is infinity, expressed as \( \lim_{{n \rightarrow \infty}} s(n) = \infty \). The provided exercise illustrates a contrasting scenario where the limit is not infinite, highlighting the importance of recognizing when infinite limits are applicable and when they're not.

Limit Properties

When solving limit problems, it’s helpful to use limit properties, which are rules that describe how limits behave under different operations. Some fundamental properties include the limit of a sum, the limit of a product, and the limit of a quotient.
For the given sequence \( s(n) = \frac{n + 1}{2n} \), we leveraged limit properties to decompose the expression into more manageable components. As \( n \) tends towards infinity, the limit of a sum, such as \( n + 1 \), is the sum of the limits of the individual terms if these limits exist.
Similarly, the limit of a quotient states that the limit of a ratio is the quotient of the limits, given that the limit of the denominator is not zero. Consequently, \( \lim_{{n \rightarrow \infty}} \frac{n+1}{2n} \) simplifies to \( \frac{1+\lim_{{n \rightarrow \infty}} \frac{1}{n}}{2} \), which further simplifies to \( \frac{1+0}{2} \) when we apply the property that the limit of a term \( \frac{1}{n} \) as \( n \) approaches infinity is zero.
These properties are integral tools in determining the limit of a sequence and demonstrate a systematic approach to solving limit problems in calculus.