Worksheets · Calculus

Alternating series and Lagrange error bound worksheet

For an alternating series whose terms shrink to zero, the error after stopping is at most the first term left out. For a Taylor polynomial, the Lagrange bound uses the largest the next derivative can be.

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Name

Error Bounds

Date Period

Answer each question.

  1. Bound the error in using the first 8 terms to approximate ∑n=1∞(−1)n+1n2
  2. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of e−x at x=13
  3. How many terms of ∑n=1∞(−1)n+1n2 guarantee an error less than 0.001?
  4. Use the Lagrange error bound to bound the error of the degree-3 Maclaurin polynomial of e−x at x=14
  5. Bound the error in using the first 5 terms to approximate ∑n=1∞(−1)n+1n
  6. Use the Lagrange error bound to bound the error of the degree-3 Maclaurin polynomial of sin⁡x at x=12
  7. How many terms of ∑n=1∞(−1)n+1n2 guarantee an error less than 0.01?
  8. Use the Lagrange error bound to bound the error of the degree-2 Maclaurin polynomial of sin⁡x at x=14
  9. How many terms of ∑n=1∞(−1)n+1n guarantee an error less than 0.001?
  10. Use the Lagrange error bound to bound the error of the degree-2 Maclaurin polynomial of sin⁡2x at x=15
  11. How many terms of ∑n=1∞(−1)n+1n2 guarantee an error less than 0.005?
  12. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of cos⁡x at x=12
  13. Bound the error in using the first 3 terms to approximate ∑n=1∞(−1)n+1n
  14. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of sin⁡2x at x=13
  15. How many terms of ∑n=1∞(−1)n+1n3 guarantee an error less than 0.001?
  16. Use the Lagrange error bound to bound the error of the degree-4 Maclaurin polynomial of cos⁡x at x=110
  17. How many terms of ∑n=1∞(−1)n+1n3 guarantee an error less than 0.005?
  18. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of e−x at x=12
  19. How many terms of ∑n=1∞(−1)n+1n guarantee an error less than 0.01?
  20. Use the Lagrange error bound to bound the error of the degree-4 Maclaurin polynomial of e−x at x=110

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Error BoundsAnswer keyVersion 1

Date Period

Answer each question.

  1. Bound the error in using the first 8 terms to approximate ∑n=1∞(−1)n+1n2at most 181
  2. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of e−x at x=13at most 1524880
  3. How many terms of ∑n=1∞(−1)n+1n2 guarantee an error less than 0.001?31
  4. Use the Lagrange error bound to bound the error of the degree-3 Maclaurin polynomial of e−x at x=14at most 16144
  5. Bound the error in using the first 5 terms to approximate ∑n=1∞(−1)n+1nat most 16
  6. Use the Lagrange error bound to bound the error of the degree-3 Maclaurin polynomial of sin⁡x at x=12at most 1384
  7. How many terms of ∑n=1∞(−1)n+1n2 guarantee an error less than 0.01?10
  8. Use the Lagrange error bound to bound the error of the degree-2 Maclaurin polynomial of sin⁡x at x=14at most 1384
  9. How many terms of ∑n=1∞(−1)n+1n guarantee an error less than 0.001?1000
  10. Use the Lagrange error bound to bound the error of the degree-2 Maclaurin polynomial of sin⁡2x at x=15at most 4375
  11. How many terms of ∑n=1∞(−1)n+1n2 guarantee an error less than 0.005?14
  12. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of cos⁡x at x=12at most 146080
  13. Bound the error in using the first 3 terms to approximate ∑n=1∞(−1)n+1nat most 14
  14. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of sin⁡2x at x=13at most 432805
  15. How many terms of ∑n=1∞(−1)n+1n3 guarantee an error less than 0.001?10
  16. Use the Lagrange error bound to bound the error of the degree-4 Maclaurin polynomial of cos⁡x at x=110at most 112000000
  17. How many terms of ∑n=1∞(−1)n+1n3 guarantee an error less than 0.005?5
  18. Use the Lagrange error bound to bound the error of the degree-5 Maclaurin polynomial of e−x at x=12at most 146080
  19. How many terms of ∑n=1∞(−1)n+1n guarantee an error less than 0.01?100
  20. Use the Lagrange error bound to bound the error of the degree-4 Maclaurin polynomial of e−x at x=110at most 112000000

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

How many terms of ∑n=1∞(−1)n+1n2 guarantee an error less than 0.01?

  1. The error is at most the first term left out, 1/(N + 1)².
  2. That is less than 0.01 when N + 1 is more than 10.
  3. So N = 10 terms.

The answer is 10.

Where students go wrong

Using the last term kept instead of the first term left out. The bound is the size of the next term.

Also called alternating series estimation theorem, Lagrange remainder, Taylor remainder or error bound.

What you can put on this worksheet

Alternating series error bound
Bound the error in using the first 8 terms to approximate ∑n=1∞(−1)n+1n2 → at most 181
Lagrange error bound
Use the Lagrange error bound to bound the error of the degree-4 Maclaurin polynomial of cos⁡x at x=15 → at most 1375000

Questions about these worksheets

Yes. Every alternating series and lagrange error bound sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the alternating series and lagrange error bound questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 2: alternating series error bound and lagrange error bound. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the last term kept instead of the first term left out. The bound is the size of the next term.