Geometric Series: Sum of x^n series = 1/(1 - x) if |x| is less than 1

https://play.3speak.tv/embed?v=mes/n4paxdaa

In this video, I show that the infinite series xⁿ = x⁰ + x¹ + x² + ... is just a geometric series, and converges if the absolute value of x is less than 1. By common convention, x⁰ = 1, so the first term a is 1, and the common ratio is x, thus the sum is given by the formula 1/(1 - x).

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#math #calculus #series #geometricseries #education

Timestamps

• Example 5: Find the sum of the series xⁿ where |x| is less than 1 – 0:00
• Solution: x⁰ is defined as equal to 1 – 0:22
• Expanding the series yields a geometric series – 1:07
• Since the common ratio x has an absolute value less than 1, the series is convergent and equals the first term a / (1 - x) – 2:02

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