Limits of Sequences: Videos & Practice Problems

Sequences are similar to functions, with discrete inputs leading to a list of outputs that often follow a pattern. When analyzing limits as the input approaches infinity, sequences can either converge to a specific value or diverge. For example, the sequence $a=\frac{1}{n}$ converges to 0, while $a=(-1){\left(}^n$ diverges, oscillating between -1 and 1. Understanding these concepts is crucial for grasping limits in calculus.

Limits of Sequences

Sequences are mathematical constructs similar to functions, where discrete inputs yield a list of outputs that often follow a specific pattern. When analyzing sequences, particularly as the input approaches infinity, we can apply the same limit rules used for functions. This approach allows us to determine whether a sequence converges or diverges.

To illustrate this, consider the sequence defined by \( a_n = \frac{1}{n} \). As \( n \) approaches infinity, the denominator increases without bound, causing the entire fraction to approach 0. Thus, we conclude that the limit of this sequence is 0, indicating that the sequence converges. In mathematical terms, we say that a sequence converges if it approaches a specific value as the input increases indefinitely.

In contrast, if we examine a sequence like \( a_n = \frac{8n + 1}{3} \), we can simplify it to observe that as \( n \) approaches infinity, the term \( 8n \) dominates, leading the sequence to increase without bound. Therefore, this sequence diverges, as it does not approach a finite limit.

Another interesting case arises with the sequence \( a_n = (-1)^n \). Here, the outputs oscillate between -1 and 1 depending on whether \( n \) is odd or even. This behavior indicates that the sequence does not settle on a single value as \( n \) increases, leading us to conclude that it diverges as well.

In summary, when analyzing sequences, we can determine convergence or divergence by examining the behavior of the sequence as \( n \) approaches infinity. A sequence converges if it approaches a specific limit, while it diverges if it does not settle on a single value. Understanding these concepts is crucial for further studies in calculus and mathematical analysis.

Evaluate $\lim_{n\to\infty}a_{n}$ and determine whether the sequence converges or diverges.


$a_{n}=5\cos\left(\frac{14}{n^2}\right)$

• A. $a_{n}=5$, converges.
• B. $a_{n}=5$, diverges.
• C. $a_{n}=DNE$, converges.
• D. $a_{n}=DNE$, diverges.

Answer: (A)

Evaluate $\lim_{n\to-\infty}a_{n}$ and determine whether the sequence converges or diverges.
$a_{n}=300\sin\left(\frac{n^2-120}{n}\right)$

• A. $a_{n}=300$, converges.
• B. $a_{n}=300$, diverges.
• C. $a_{n}=DNE$, converges.
• D. $a_{n}=DNE$, diverges.

Answer: (D)

Here’s what students ask on this topic:

What is the limit of the sequence a_n = 1/n as n approaches infinity?

The limit of the sequence a_n = 1/n as n approaches infinity is 0. As n becomes very large, the value of 1/n becomes very small, approaching 0. This indicates that the sequence converges to 0. In mathematical terms, we express this as lim_n→∞ (1/n) = 0. This behavior is typical for sequences where the term involves a fraction with n in the denominator and no n in the numerator, leading to convergence to zero.

How do you determine if a sequence converges or diverges?

To determine if a sequence converges or diverges, analyze the behavior of its terms as n approaches infinity. If the terms approach a specific finite value, the sequence converges to that value. For example, a_n = 1/n converges to 0. If the terms do not approach a specific value, the sequence diverges. For instance, a_n = (8n + 1)/3 diverges as it increases without bound. Additionally, sequences like a_n = (-1)ⁿ oscillate between values and do not converge, thus they diverge. Simplifying expressions and creating tables of values can help observe these trends.

What does it mean for a sequence to oscillate?

A sequence oscillates when its terms alternate between different values without settling on a single limit as n approaches infinity. This means the sequence does not converge to a specific value. An example is the sequence a_n = (-1)ⁿ, which alternates between -1 and 1. Such sequences are considered divergent because they do not approach a single finite value. Oscillation is a common behavior in sequences involving alternating signs or periodic functions.

How can you use a table to determine the limit of a sequence?

Using a table to determine the limit of a sequence involves calculating and listing the sequence's terms for increasing values of n. By observing the trend of these terms, you can infer whether the sequence converges or diverges. For example, for a_n = 1/n, as n increases, the terms decrease towards 0, indicating convergence to 0. If the terms do not approach a specific value or oscillate, the sequence diverges. This method provides a visual representation of the sequence's behavior as n approaches infinity.

What is the difference between convergence and divergence in sequences?

Convergence in sequences occurs when the terms approach a specific finite value as n approaches infinity. For example, a_n = 1/n converges to 0. Divergence occurs when the terms do not approach a specific value, either increasing without bound or oscillating. For instance, a_n = (8n + 1)/3 diverges as it increases indefinitely, and a_n = (-1)ⁿ diverges due to oscillation. Convergence indicates a stable limit, while divergence indicates instability or unbounded behavior.