Ratio Test, Infinite Series, Convergence, Divergence, Examples - Calculus

The ratio test determines if an infinite series converges or diverges by examining the limit of the ratio of consecutive terms, \(|a_{n+1}/a_{n}|\). If the limit \(L\) is less than 1, the series converges absolutely; if \(L\) is greater than 1 (or diverges to infinity), the series diverges; and if \(L\) equals 1, the test is inconclusive. This test is particularly useful for series involving factorials and powers.

💡How to apply the ratio test
• Set up the ratio: Form the ratio of the \((n+1)^{th}\) term to the \(n^{th}\) term of the series, \(|a_{n+1}|/|a_{n}|\).
• Calculate the limit: Find the limit of this ratio as \(n\) approaches infinity. Let this limit be \(L\).
• Interpret the result:
◦ If \(L lt 1\), the series converges absolutely.
◦ If \(L gt 1\) (or if the limit is infinity), the series diverges.
◦ If \(L=1\), the ratio test is inconclusive, and you must use another convergence test.

💡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 Ratio test
01:20 Worked example

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