Find the value of an infinite geometric series

Learning Objectives Define an infinite geometric series and its components. Identify the first term (a₁) and the common ratio (r) of an infinite geometric series. Determine whether an infinite geometric series converges or diverges based on the value of its common ratio. Apply the formula S = a₁ / (1 - r) to calculate the sum of a convergent infinite geometric series. Convert a repeating decimal into a fraction by modeling it as an infinite geometric series. Solve real-world problems involving the sum of an infinite geometric series. Explain why a finite sum only exists when the absolute value of the common ratio is less than 1. Imagine a bouncing ball that always rebounds to 2/3 of its previous height. Will it ever stop bouncing? 🏀 Let's explore the infinite math be...

TermDefinitionExample Geometric SeriesThe sum of the terms of a geometric sequence. A geometric sequence is one where each term is found by multiplying the previous term by a constant value.The sequence 4, 8, 16, 32... gives the series 4 + 8 + 16 + 32 + ... Common Ratio (r)The constant value multiplied to get from one term to the next in a geometric sequence. It is found by dividing any term by its preceding term (r = a₂ / a₁).In the series 27 + 9 + 3 + ..., the common ratio is r = 9 / 27 = 1/3. Infinite Geometric SeriesA geometric series that continues without end; it has an infinite number of terms.1 + 1/2 + 1/4 + 1/8 + ... (the '...' indicates it goes on forever). ConvergenceAn infinite series converges if its sequence of partial sums approaches a specific, finite number as t...

Condition for Convergence |r| < 1 (which is the same as -1 < r < 1) This is the most important rule. You must check this first. An infinite geometric series only has a finite sum if the absolute value of its common ratio is less than 1. If this condition is not met, the series diverges and has no sum. Sum to Infinity Formula S = a₁ / (1 - r) Use this formula to calculate the sum (S) of a convergent infinite geometric series. Here, 'a₁' is the first term of the series and 'r' is the common ratio. This formula ONLY works if |r| < 1. Condition for Divergence |r| ≥ 1 If the absolute value of the common ratio is greater than or equal to 1, the series diverges. Do not attempt to use the sum formula, as the sum is not a finite number.