What is a Sequence in Calculus? Definition & Examples

📚 What is a Sequence in Calculus?

In calculus, a sequence is an ordered list of numbers. It can be finite or infinite. Each number in the sequence is called a term, and the terms are usually indexed by positive integers.

Quick Study Guide

• 🔢 A sequence is a function whose domain is the set of natural numbers.
• ♾️ An infinite sequence continues indefinitely.
• 📉 A sequence can converge to a limit, meaning its terms get closer and closer to a specific value.
• 📈 A sequence can diverge if it doesn't approach a specific limit.
• ➕ Arithmetic sequences have a constant difference between consecutive terms.
• ✖️ Geometric sequences have a constant ratio between consecutive terms.
• 📝 Common Notation: ${a_1, a_2, a_3, ..., a_n, ...}$
• 🧮 Limit of a Sequence: $\lim_{n \to \infty} a_n = L$

Practice Quiz

1. Which of the following defines a sequence in calculus?
   1. A set of unordered numbers.
   2. A function whose domain is the set of real numbers.
   3. A function whose domain is the set of natural numbers.
   4. A set of complex numbers.
2. What does it mean for a sequence to converge?
   1. The terms increase without bound.
   2. The terms decrease without bound.
   3. The terms approach a specific value.
   4. The terms oscillate between two values.
3. Which of the following is an example of an arithmetic sequence?
   1. 2, 4, 8, 16, ...
   2. 1, 1, 2, 3, 5, ...
   3. 1, 4, 9, 16, ...
   4. 3, 6, 9, 12, ...
4. What is the common ratio in the geometric sequence 2, 6, 18, 54, ...?
   1. 2
   2. 3
   3. 4
   4. 6
5. If $\lim_{n \to \infty} a_n = L$, what does L represent?
   1. The sum of the sequence.
   2. The limit of the sequence.
   3. The derivative of the sequence.
   4. The integral of the sequence.
6. Which sequence is defined by the formula $a_n = n^2$?
   1. 1, 3, 5, 7, ...
   2. 1, 2, 3, 4, ...
   3. 1, 4, 9, 16, ...
   4. 2, 4, 6, 8, ...
7. What is a key difference between a sequence and a series?
   1. A sequence is a sum, and a series is a list.
   2. A sequence is a list, and a series is a sum.
   3. A sequence is always infinite, and a series is always finite.
   4. A sequence and series are the same thing.