Convergent and divergent geometric series

Learning Objectives Identify the first term (a) and the common ratio (r) of an infinite geometric series. Determine whether an infinite geometric series is convergent or divergent by analyzing its common ratio. Calculate the sum of a convergent infinite geometric series using the appropriate formula. Explain why a divergent infinite geometric series does not have a finite sum. Convert a repeating decimal into a fraction using the concept of a convergent geometric series. Solve real-world problems involving convergent geometric series, such as calculating the total distance traveled by a bouncing ball. If a bouncing ball always rebounds to 3/4 of its previous height, will it ever stop bouncing? 🎾 Let's find out if the total distance it travels is finite or infinite! In...

TermDefinitionExample Infinite Geometric SeriesThe sum of the terms of a geometric sequence that continues forever. It is written in the form a + ar + ar^2 + ar^3 + ...The series 16 + 8 + 4 + 2 + ... is an infinite geometric series where a = 16 and r = 1/2. Common Ratio (r)The constant factor you multiply by to get from one term to the next in a geometric sequence or series.In the series 100 + 20 + 4 + ..., the common ratio is r = 20/100 = 1/5. Convergent SeriesAn infinite series whose partial sums approach a specific, finite number as the number of terms increases. This finite number is called the 'sum to infinity'.The series 1 + 1/2 + 1/4 + 1/8 + ... converges to a sum of 2. Divergent SeriesAn infinite series whose partial sums do not approach a finite number. The sum grows in...

Condition for Convergence |r| < 1 An infinite geometric series converges (has a finite sum) if and only if the absolute value of its common ratio 'r' is less than 1. This means -1 < r < 1. Sum to Infinity Formula S_\infty = \frac{a}{1 - r} Use this formula to calculate the sum of a convergent infinite geometric series, where 'a' is the first term and 'r' is the common ratio. This formula ONLY works if |r| < 1. Condition for Divergence |r| \ge 1 An infinite geometric series diverges (does not have a finite sum) if the absolute value of its common ratio 'r' is greater than or equal to 1.