What Are Sequences and Series?

A sequence is an ordered list of numbers, like \(1, 2, 3, 4, \ldots\). A series is what you get when you add the terms of a sequence together, such as \(1 + 2 + 3 + 4 + \ldots\). In calculus, we explore how to find sums for infinite series and determine if those sums even exist!

Arithmetic and Geometric Sequences

• Arithmetic sequence: Each term increases by the same amount.
  Formula for the \(n\)th term: \(a_n = a_1 + (n-1)d\)
• Geometric sequence: Each term is multiplied by the same value.
  Formula for the \(n\)th term: \(a_n = a_1 \cdot r^{n-1}\)

Infinite Series

Some series keep going forever! Calculus helps us decide if these infinite sums have a limit (they "converge") or just keep growing (they "diverge").

Why Study Sequences and Series?

Sequences and series are everywhere: from calculating interest in banks to analyzing signals in engineering. Understanding them is essential for higher-level math, science, and real-world problem-solving.