Problem 39 - Calculus for Biology and Medicine

Question

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{n}{n+1} $$

Step-by-Step Solution

Answer

The first five terms are 0, \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{3}{4}\), and \(\frac{4}{5}\). The limit is 1.

Step 1: Calculate First Term

Substitute \(n = 0\) into the sequence formula to find \(a_0\). \[ a_0 = \frac{0}{0+1} = \frac{0}{1} = 0 \] Thus, the first term is 0.

Step 2: Calculate Second Term

Substitute \(n = 1\) into the sequence formula to find \(a_1\). \[ a_1 = \frac{1}{1+1} = \frac{1}{2} \]Thus, the second term is \(\frac{1}{2}\).

Step 3: Calculate Third Term

Substitute \(n = 2\) into the sequence formula to find \(a_2\). \[ a_2 = \frac{2}{2+1} = \frac{2}{3} \]Thus, the third term is \(\frac{2}{3}\).

Step 4: Calculate Fourth Term

Substitute \(n = 3\) into the sequence formula to find \(a_3\). \[ a_3 = \frac{3}{3+1} = \frac{3}{4} \]Thus, the fourth term is \(\frac{3}{4}\).

Step 5: Calculate Fifth Term

Substitute \(n = 4\) into the sequence formula to find \(a_4\). \[ a_4 = \frac{4}{4+1} = \frac{4}{5} \]Thus, the fifth term is \(\frac{4}{5}\).

Step 6: Derive the Limit

To find the limit of \(a_n\) as \(n\) approaches infinity, consider the expression:\[ a_n = \frac{n}{n+1} = \frac{n}{n} \cdot \frac{1}{1+\frac{1}{n}} \]As \(n\) becomes very large, \(\frac{1}{n}\) becomes negligible.Thus, \(a_n \approx \frac{1}{1} = 1\).So, the limit \(\lim_{n \to \infty} a_n = 1\).

Key Concepts

Calculus Sequences

Calculus provides a framework for understanding changes and behaviors within sequences and series. A sequence in calculus is a set of numbers ordered in a linear fashion such as \(a_0, a_1, a_2, \ldots \). Each number in the sequence is called a term. Sequences could be finite or infinite. In this context, we are dealing with a sequence defined by a formula, \(a_n = \frac{n}{n+1}\). This formula dictates how each term \(a_n\) is related to its position \(n\) within the sequence. To better understand sequences, we calculate specific terms:

• For \(n = 0\), \(a_0 = 0\)
• For \(n = 1\), \(a_1 = \frac{1}{2}\)
• For \(n = 2\), \(a_2 = \frac{2}{3}\)
• For \(n = 3\), \(a_3 = \frac{3}{4}\)
• For \(n = 4\), \(a_4 = \frac{4}{5}\)

By calculating these terms, we can observe that as \(n\) increases, \(a_n\) approaches a particular behavior, leading us to investigate the concept of limits.

Infinite Limits

Understanding infinite limits is integral in evaluating the behavior of sequences as their indices grow indefinitely. The limit of a sequence \(\lim_{n \to \infty} a_n\) essentially describes what value \(a_n\) approaches as \(n\) becomes very large. For the sequence \(a_n = \frac{n}{n+1}\), finding the limit involves simplifying the expression:

Notice that the major change as \(n\) increases is the effect of the \(\frac{1}{n+1}\) term becoming very small (or negligible). We approximate:
\[ a_n = \frac{n}{n+1} = \frac{n}{n} \cdot \frac{1}{1+\frac{1}{n}} \]

Since \(\frac{1}{n}\) approaches 0 as \(n\) approaches infinity, the expression simplifies:
\[ \frac{1}{1+\frac{1}{n}} \approx 1 \]

Thus \(a_n \approx 1\), leading us to conclude that the infinite limit of this sequence is 1: \(\lim_{n \to \infty} a_n = 1\). Understanding infinite limits help us grasp how sequences behave at the "ends" of their progressions.

Convergence of Sequences

In calculus, a sequence is said to converge if it approaches a specific value as the index (\(n\)) goes to infinity. This destination value is known as the limit of the sequence. If the sequence \(a_n = \frac{n}{n+1}\) converges, it means that the terms settle towards a particular number. This behavior is crucial in understanding the long-term impact or stability of scenarios modeled by sequences.

Convergence can be visualized:

• The terms: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\) are getting closer to 1 as they progress.
• The closer the terms get to 1, the slower their rate of change becomes.
• Eventually, as \(n\) reaches very high values, \(a_n\) resides near 1.

For sequences that converge, this limit is a stabilizing factor indicating where the sequence "rests." Convergence is a fundamental concept in calculus used for understanding how mathematical models predict steadiness or persistence over time.