Find the value of an infinite geometric series

Learning Objectives Define an infinite geometric series and its components. Identify the first term (a) and the common ratio (r) for any given geometric series. Determine whether an infinite geometric series converges or diverges by analyzing its common ratio. Apply the formula S = a / (1 - r) to calculate the sum of a convergent infinite geometric series. Solve problems involving infinite geometric series presented in expanded form or sigma notation. Convert a repeating decimal into a fraction by modeling it as an infinite geometric series. Imagine a bouncing ball that always returns to 2/3 of its previous height. If it's dropped from 9 meters, will it travel a finite distance or bounce forever? 🏀 Let's find out! This tutorial explores the fascinating concept of...

TermDefinitionExample Geometric SeriesThe sum of the terms of a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant value called the common ratio.The sequence 3, 6, 12, 24... gives the series 3 + 6 + 12 + 24 + ... Common Ratio (r)The fixed, non-zero number you multiply by to get from one term to the next in a geometric sequence or series.In the series 100 + 50 + 25 + ..., the common ratio is r = 50/100 = 0.5. Partial Sum (S_n)The sum of the first 'n' terms of a series. The value of an infinite series is the limit of its sequence of partial sums as n approaches infinity.For the series 1 + 1/2 + 1/4 + ..., the third partial sum is S_3 = 1 + 1/2 + 1/4 = 1.75. ConvergenceAn infinite series is said to converge if its sequence...

Condition for Convergence An infinite geometric series converges if and only if |r| < 1 (or -1 < r < 1). This is the first and most important test. Before attempting to find a sum, you must calculate the common ratio 'r' and check if its absolute value is less than 1. If |r| ≥ 1, the series diverges and has no finite sum. Sum of a Convergent Infinite Geometric Series S = \frac{a}{1 - r} If the series converges (i.e., |r| < 1), this formula is used to find its sum. 'S' represents the sum, 'a' is the first term of the series, and 'r' is the common ratio.