Question

In Exercises 81-84, give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) A monotonically increasing sequence that converges to 10

Short answer

The sequence \(a_n = 10 - 1/n\) is an example of a monotonically increasing sequence that converges to 10. As the value of \(n\) gets larger, the terms of the sequence become larger as well and they draw nearer to the limit 10.

Step-by-step solution

Understanding Monotonically Increasing Sequence

Monotonically increasing sequence means that each term in the sequence is greater than or equal to the previous term. This might be accomplished by adding a certain value to the previous term.

Understanding Convergence to a Limit

A sequence converges to a limit (in this case 10), if the terms in the sequence become arbitrarily close to that limit as the sequence progresses. So, for the sequence to converge to 10, the difference between the term and 10 should become smaller as the sequence progresses.

Creating a Formula

An example of a monotonically increasing sequence that converges to 10 can be represented as \(a_n = 10 - 1/n\). This sequence fits the requirements because each term \(a_{n}\) is greater than the previous term \(a_{n-1}\), and it tends towards limit 10 as \(n\) extends to infinity. To illustrate this, lets consider a few terms of this sequence \(a_1 = 9, a_2 = 9.5, a_3 = 9.67, a_4 = 9.75 ... \). As you can see each term is larger than the previous and as \(n\) extends to infinity, the terms become very close to 10.

Key concepts

Convergence to a Limit

Understanding the concept of convergence to a limit is crucial when studying sequences in mathematics. Convergence refers to the behavior of a sequence whereby its terms become increasingly close to a specific value, known as the limit, as the sequence progresses indefinitely. A monotonically increasing sequence that converges has terms that grow gradually larger, yet they will never exceed a certain bounding value.

Take, for instance, the sequence given by the formula \(a_n = 10 - \frac{1}{n}\). As \(n\) increases, the fraction \(\frac{1}{n}\) gets smaller, implying that \(a_n\) approaches 10 more closely with each subsequent term. For a sequence to converge to a limit, like 10 in this example, one can think of the terms as a series of steps on a ladder reaching up to a ledge. As the steps (terms) go higher (increase in value), they get closer to the ledge (the limit), but crucially, without ever stepping over the ledge.

This concept is a cornerstone of calculus and analysis since it lays the groundwork for understanding series, functions, and integrals which are all vital in the study of continuous mathematical phenomena.

Sequence in Mathematics

A sequence in mathematics is an ordered list of numbers, typically following a specific pattern or rule. These numbers are called the 'terms' of the sequence. The sequence's rule determines whether it is increasing, decreasing, constant, or otherwise. An increasing sequence, like the one in our example, includes terms that never decrease as they progress from one to the next.

In the context of monotonically increasing sequences, the term 'monotonic' indicates that the sequence never descends. It either ascends or remains flat as it advances from each term to the next. Monotonicity can highlight a sequence's predictable behavior and is often considered when assessing convergence properties or bounding the values of sequences in more complex mathematical problems. Knowing the nature of the sequence is a fundamental aspect not only in solving textbook problems but also in real-world applications where sequences can model growth patterns and trends over time.

Infinite Series

The concept of an infinite series is closely related to sequences and convergence. While a sequence is a list of terms, once we start talking about adding those terms together, we enter the realm of series. An infinite series is simply the sum of an infinite sequence of numbers.

One common question in mathematics is whether an infinite series converges, which means its partial sums approach a finite limit. Understanding whether an infinite series converges is essential since it can represent a finite quantity despite being a never-ending process, like the geometric progression of fractions that get infinitely small but add up to a concrete number.

For example, the infinite series formed from the sequence \(a_n = 10 - \frac{1}{n}\) would be examined by adding all of its terms together - a task that requires different techniques than those used for understanding sequences. Infinite series are particularly interesting in theories of integration, probability, and are integral to the underpinnings of multiple areas such as physics and economics.