Alternating Series Test, Infinite Series, Conditions, Examples and Solutions - Calculus

The alternating series test determines if an alternating series converges by checking two conditions: the absolute value of its terms must be a decreasing sequence, and the limit of these terms as \(n\) approaches infinity must be zero. If both conditions are met, the series converges.

💡Conditions for the alternating series test
• Terms are decreasing: The sequence \(b_{n}\) must be decreasing, meaning \(b_{n+1}\le b_{n}\) for all \(n\) greater than or equal to some integer \(N\).
• Limit of terms is zero: The limit of the sequence \(b_{n}\) as \(n\) approaches infinity must be zero (\(\lim _{n\rightarrow \infty }b_{n}=0\)). This is also known as the first part of the {divergence test}.

💡How to apply the test
• Identify the non-alternating part of the series: For an alternating series like \(\sum (-1)^{n+1}b_{n}\) or \(\sum (-1)^{n}b_{n}\), identify the positive sequence \(b_{n}\).
• Check the limit: Calculate the limit of \(b_{n}\) as \(n\) approaches infinity. If this limit is not zero, the series diverges by the divergence test, and the alternating series test cannot be used to prove convergence.
• Check if the sequence is decreasing: Show that \(b_{n+1}\le b_{n}\) for all large enough \(n\). This can often be done by comparing the terms directly or by using calculus techniques like finding the derivative of the function corresponding to \(b_{n}\).
• Conclude: If both conditions (decreasing terms and the limit of the terms is zero) are met, the alternating series converges. If the conditions are not met, the test is inconclusive and cannot be used to prove convergence.

💡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 Alternating series test
01:03 Worked example

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