Adding Up Forever: Sequences and Series

Sometimes, we want to add up infinitely many numbers—like calculating compound interest over time or analyzing population growth.

Sequences

A sequence is an ordered list of numbers, like \( 1, 1/2, 1/4, 1/8, \ldots \).

Series and Convergence

A series is the sum of a sequence. Not all series add up to a finite number—if they do, we say the series “converges.”

• Geometric series: Adds terms like \( a, ar, ar^2, \ldots \)
• Convergence tests: Help us see if a series really adds up to something finite.

Why It Matters

Series help model real-world phenomena like finance, physics, and biology.