Divergent Geometric Series?

📌 CONCEPT: A divergent geometric series is a series that does not converge to a finite sum, meaning that the sum of the series will grow indefinitely as more terms are added.

📐 RULE / FORMULA: A geometric series is divergent if its common ratio |r| ≥ 1, otherwise it is convergent.

💡 WORKED EXAMPLE: Consider a geometric series with first term a = 1 and common ratio r = -1/2. We have |r| = |-1/2| = 1/2 < 1, so the series will converge. However, if the common ratio were r = 1 or r = -1, the series would be divergent.

⚠️ COMMON MISTAKE: Students often mistakenly assume that a geometric series will converge if the common ratio is less than 1, without considering the case where the common ratio is 1 or -1, which would result in a divergent series.