Question

Consider the formulas for the following sequences. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. $$a_{n}=\frac{100 n-1}{10 n} ; n=1,2,3, \ldots$$

Short answer

Based on the given sequence formula \(a_n=\frac{100n-1}{10n}\) and after observing the pattern in the first 10 terms, we determined that the limit of the sequence exists and is equal to 10 as \(n\) approaches infinity.

Step-by-step solution

Identify the given sequence formula

We are given the formula for the sequence: $$a_n=\frac{100n-1}{10n}; n=1,2,3,\ldots$$

Generate the table with 10 terms

We will calculate the first 10 terms by replacing \(n\) in the formula by the numbers 1, 2, 3, ...., 10. The terms of the sequence will be presented in a table format. | \(n\) | \(a_n\) | |-----|-----------------------------------------| | 1 | \(\frac{100(1)-1}{10(1)} = \frac{99}{10}\) | | 2 | \(\frac{100(2)-1}{10(2)} = \frac{199}{20}\) | | 3 | \(\frac{100(3)-1}{10(3)} = \frac{299}{30}\) | | 4 | \(\frac{100(4)-1}{10(4)} = \frac{399}{40}\) | | 5 | \(\frac{100(5)-1}{10(5)} = \frac{499}{50}\) | | 6 | \(\frac{100(6)-1}{10(6)} = \frac{599}{60}\) | | 7 | \(\frac{100(7)-1}{10(7)} = \frac{699}{70}\) | | 8 | \(\frac{100(8)-1}{10(8)} = \frac{799}{80}\) | | 9 | \(\frac{100(9)-1}{10(9)} = \frac{899}{90}\) | | 10 | \(\frac{100(10)-1}{10(10)} = \frac{999}{100}\) |

Observe the pattern and find a plausible limit

Looking at the table, we observe that the terms of the sequence are getting closer to a certain value. We can use the concept of limits to determine if the sequence converges to a specific value as \(n\) approaches infinity. By dividing both the numerator and the denominator of \(\frac{100n-1}{10n}\) by \(n\), we can simplify the expression: $$a_n=\frac{100-\frac{1}{n}}{10}$$ Now, as \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0: $$\lim_{n\to\infty}\frac{100-\frac{1}{n}}{10}=\frac{100-0}{10}=\frac{100}{10}=10.$$ Hence, it is plausible that the limit of the sequence exists and is equal to 10.

Key concepts

Sequence Formulas

A **sequence formula** is a mathematical expression that defines the terms in a sequence. Each term in a sequence can be calculated from this formula. For example, consider the sequence formula given by \(a_n = \frac{100n-1}{10n}\). Here, \(n\) represents the position of the term in the sequence, and each term \(a_n\) is calculated based on this position.

Using such formulas, you can generate numerous terms without manually computing each one, saving time and effort. These formulas often reveal patterns and help in understanding how sequences evolve as \(n\) increases. In our case, calculating the first 10 terms using the formula can confirm the trend and behavior of the sequence.

• **Identify Variables**: \(n\) is the position of the term, while \(a_n\) is the term at that position.
• **Plug in Values**: Substitute consecutive integers for \(n\) to find \(a_1, a_2, a_3,\) and so on.
• **Observe Patterns**: Look for common behaviors among terms, such as consistent increase or decrease.

Convergence

The concept of **convergence** in sequences refers to the behavior of a sequence's terms as they progress towards a specific value as \(n\) approaches infinity. If a sequence is said to converge, it means that the terms are getting closer and closer to a particular number.

To determine convergence, observe the sequence's behavior over a series of terms. For instance, with the formula \(a_n = \frac{100n-1}{10n}\), simplifying it to \(a_n = \frac{100 - \frac{1}{n}}{10}\) gives a clearer idea. As \(n\) becomes larger, the fraction \(\frac{1}{n}\) becomes smaller, eventually tending towards 0. This means \(a_n\) approaches \(\frac{100}{10} = 10\). Thus, we say the sequence converges to 10.

• **Analyze Simplified Expressions**: Simplifying terms often makes identifying the behavior easier.
• **Observe Long Term Behavior**: Convergence is about what happens as \(n\) goes to infinity.
• **Check for Trends**: Consistent approaching towards a specific number indicates convergence.

Infinite Sequences

**Infinite sequences** are sequences that continue indefinitely. Unlike a finite sequence, which has a specific number of terms, an infinite sequence does not have a last term.

Consider the sequence defined by \(a_n = \frac{100n-1}{10n}\), which extends as long as you can substitute larger values for \(n\). Most importantly, infinite sequences allow us to study behaviors such as convergence and divergence over a long span. By examining the terms with increasing \(n\), we gain insight into the sequence's overall trend.

• **No End Point**: Infinite sequences continue without a fixed final term.
• **Useful for Analysis**: They help identify convergence, divergence, or cyclical patterns in mathematical sequences.
• **Theoretical Foundation**: Infinite sequences form the basis for more advanced concepts like calculus and series.