A Converging Series?

📌 CONCEPT: A series is convergent if the sum of its terms approaches a finite limit as the number of terms increases without bound.

📐 RULE / FORMULA: We use the formula for the sum of an infinite geometric progression (GP), S = ¯(a) / (1 - r), where 'a' is the first term and 'r' is the common ratio.

💡 WORKED EXAMPLE: Given the series 1 - 1/2 + 1/4 - 1/8 + ..., we can see that it is a geometric progression with first term 'a' = 1 and common ratio 'r' = -1/2. Since |r| < 1, the series is convergent. Using the formula for sum of infinite GP, S = ¯(1) / (1 + 1/2) = 2/3.

⚠️ COMMON MISTAKE: Students might incorrectly assume that a series is always divergent if the common ratio is greater than 1. However, the condition |r| < 1 is necessary for the series to be convergent, but not sufficient if r = 1 or r = -1, as the series may still be divergent in those cases.