Geometric Series, Partial Sum, Convergence, Example - Calculus

A geometric series converges if the absolute value of its common ratio, \(r\), is less than 1. If \(|r|\ge 1\), the series diverges, meaning the sum does not approach a finite value. If a geometric series converges, its sum can be calculated using the formula \(\frac{a}{1-r}\), where 'a' is the first term and 'r' is the common ratio.

💡Convergence
• Condition for convergence: The series converges if \(|r| lt 1\). This is the same as saying the common ratio \(r\) must be between -1 and 1.
• Condition for divergence: The series diverges if \(|r|\ge 1\), meaning \(r\le -1\) or \(r\ge 1\).
• Sum of a convergent series: If the series converges, its infinite sum is \(\frac{a}{1-r}\), where \(a\) is the first term of the series and \(r\) is the common ratio.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1DlYi7xcmyEAhWTnwUN8zRgisBdHIdmtr/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1VEHFVDs67vvvQVVsozjmOCO5RbEgzbkA/view?usp=drive_link

💡Chapters:
00:00 Geometric series, definition, sum, convergence
02:20 Worked example

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