Find the value of an infinite geometric series

Learning Objectives Define a geometric sequence and identify its common ratio. Recognize an infinite geometric series as a sum of terms that continues indefinitely. Determine if the terms in an infinite geometric series are getting smaller. Understand that an infinite geometric series can have a finite 'value' or total sum if its terms get smaller. Use visual models to understand how an infinite series can approach a whole. Explain why some infinite series do not have a finite value. Have you ever thought about adding numbers forever? 🤔 What if the numbers get smaller and smaller each time? Can you still get a total sum? In this lesson, we'll explore special number patterns called geometric series that go on forever! We'll discover how to find out if th...

TermDefinitionExample SequenceAn ordered list of numbers, like 2, 4, 6, 8, ...The sequence of even numbers: 2, 4, 6, 8, 10. Geometric SequenceA sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.100, 50, 25, 12.5, ... (each term is half of the previous one). Common Ratio (r)The fixed number you multiply by to get from one term to the next in a geometric sequence.In the sequence 3, 6, 12, 24, ..., the common ratio (r) is 2 because 3 × 2 = 6, 6 × 2 = 12, and so on. SeriesThe sum of the terms of a sequence.The series for 1, 2, 3, 4 is 1 + 2 + 3 + 4 = 10. Infinite SeriesA series that continues without end; it has an infinite number of terms.1 + 1/2 + 1/4 + 1/8 + ... (the '...' means it goes on forever). Converging SeriesAn i...

Finding the Common Ratio $r = \frac{\text{Any Term}}{\text{Previous Term}}$ To find the common ratio (r) in a geometric sequence, divide any term by the term that came just before it. This ratio tells you what number is being multiplied repeatedly. Condition for a Finite Infinite Sum $0 < r < 1$ An infinite geometric series will have a specific, finite total value (it 'converges') only if its common ratio, $r$, is a positive number smaller than 1. This means the terms are always getting smaller. If $r$ is 1 or larger, the sum will grow infinitely.