Alternating Series, Error Bound, Estimate, Practice Problems - Calculus

The error in approximating an alternating series with a partial sum is the difference between the exact sum and the approximation, also known as the remainder. The alternating series error bound states that the absolute value of this error is less than or equal to the absolute value of the first neglected term. For an alternating series that converges and whose terms decrease in absolute value, the error (\(|R_{N}|\)) is less than or equal to the absolute value of the \((N+1)\)-th term (\(a_{N+1}\)).

💡Key concepts
• Partial sum (\(S_{N}\)): The sum of the first \(N\) terms of a series.
• Remainder (\(R_{N}\)): The difference between the infinite sum (\(S\)) and the partial sum (\(S_{N}\)). This is the error: \(R_{N}=S-S_{N}\).
• Error Bound: An upper limit on the magnitude of the error, \(|R_{N}|\).

💡How to find the error bound
• Check for convergence: First, confirm that the alternating series converges using the Alternating Series Test. This requires that the absolute value of the terms are decreasing and approach zero.
• Identify the first neglected term: Determine the first term that is not included in your partial sum. If you are using the first \(N\) terms, this is the \((N+1)\)-th term, denoted as \(a_{N+1}\).
• Apply the formula: The error bound is the absolute value of this next term.
◦ \(|R_{N}|\le |a_{N+1}|\)
◦\(|R_{N}|\) is the absolute value of the error (or remainder) after summing \(N\) terms.
◦\(|a_{N+1}|\) is the absolute value of the first term you left out.

💡Example
• Series: \(1-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}+\dots \)
• Partial sum: Use the first four terms (\(S_{4}=1-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}=\frac{115}{144}\)).
• First neglected term: The fifth term is \(a_{5}=\frac{1}{25}\).
• Error: The error is the difference between the true sum and \(S_{4}\). The error bound is the absolute value of the first neglected term.
• Bound: \(|R_{4}|\le |\frac{1}{25}|\), so the error is no more than \(\frac{1}{25}\). This means the actual sum is within \(\frac{1}{25}\) of the partial sum \(\frac{115}{144}\).

💡Why it's important
• Rearrangement: A key advantage of absolutely convergent series is that the order of the terms can be rearranged without changing the sum. This is not true for conditionally convergent series, which can be rearranged to sum to any value.
• Testing tools: Tools like the ratio test can be used to test for absolute convergence. If the test is passed, the series is absolutely convergent and therefore also convergent.

💡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 error bound
00:51 Worked example

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