Worksheets · Calculus

Absolute and conditional convergence worksheet

A series converges absolutely when it still converges with every sign made positive. If it only converges because the signs alternate and cancel, it converges conditionally. And if its terms do not go to zero, it diverges either way.

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Sigma Prep · sigmaprep.io/worksheets

Name

Absolute and Conditional Convergence

Date Period

Decide whether each series converges absolutely, converges conditionally, or diverges, or find the values of k asked for.

  1. ∑n=1∞(−1)n6nn!
  2. For what values of k does ∑n=1∞(−1)nnkn9 converge absolutely, and conditionally?
  3. ∑n=1∞(−1)n+1n+7
  4. For what values of k does ∑n=1∞(−1)nnk+3 converge absolutely, and conditionally?
  5. ∑n=1∞(−1)nln⁡(n+2)
  6. For what values of k does ∑n=1∞(−1)nn2k converge absolutely, and conditionally?
  7. ∑n=1∞(−1)n+1nn2+7
  8. For what values of k does ∑n=1∞(−1)nn2k−5 converge absolutely, and conditionally?
  9. ∑n=1∞(−1)nn4
  10. For what values of k does ∑n=1∞(−1)nnk−2 converge absolutely, and conditionally?
  11. ∑n=1∞(−1)nn35n
  12. For what values of k does ∑n=1∞(−1)nnkn7 converge absolutely, and conditionally?
  13. ∑n=1∞(−1)n+1n2n+2
  14. For what values of k does ∑n=1∞(−1)nn2k+1 converge absolutely, and conditionally?
  15. ∑n=1∞(−1)n−1n2
  16. For what values of k does ∑n=1∞(−1)nnk+5 converge absolutely, and conditionally?
  17. ∑n=1∞(−1)n+1n
  18. For what values of k does ∑n=1∞(−1)nnkn4 converge absolutely, and conditionally?
  19. ∑n=1∞(−1)n+1ln⁡(n+3)
  20. For what values of k does ∑n=1∞(−1)nn2k−4 converge absolutely, and conditionally?

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

Sigma Prep · sigmaprep.io/worksheets

Name

Absolute and Conditional ConvergenceAnswer keyVersion 1

Date Period

Decide whether each series converges absolutely, converges conditionally, or diverges, or find the values of k asked for.

  1. ∑n=1∞(−1)n6nn!Converges absolutely
  2. For what values of k does ∑n=1∞(−1)nnkn9 converge absolutely, and conditionally?Absolutely: k<8Conditionally: 8≤k<9
  3. ∑n=1∞(−1)n+1n+7Converges conditionally
  4. For what values of k does ∑n=1∞(−1)nnk+3 converge absolutely, and conditionally?Absolutely: k>−2Conditionally: −3<k≤−2
  5. ∑n=1∞(−1)nln⁡(n+2)Converges conditionally
  6. For what values of k does ∑n=1∞(−1)nn2k converge absolutely, and conditionally?Absolutely: k>12Conditionally: 0<k≤12
  7. ∑n=1∞(−1)n+1nn2+7Converges conditionally
  8. For what values of k does ∑n=1∞(−1)nn2k−5 converge absolutely, and conditionally?Absolutely: k>3Conditionally: 52<k≤3
  9. ∑n=1∞(−1)nn4Converges absolutely
  10. For what values of k does ∑n=1∞(−1)nnk−2 converge absolutely, and conditionally?Absolutely: k>3Conditionally: 2<k≤3
  11. ∑n=1∞(−1)nn35nConverges absolutely
  12. For what values of k does ∑n=1∞(−1)nnkn7 converge absolutely, and conditionally?Absolutely: k<6Conditionally: 6≤k<7
  13. ∑n=1∞(−1)n+1n2n+2Diverges
  14. For what values of k does ∑n=1∞(−1)nn2k+1 converge absolutely, and conditionally?Absolutely: k>0Conditionally: −12<k≤0
  15. ∑n=1∞(−1)n−1n2Converges absolutely
  16. For what values of k does ∑n=1∞(−1)nnk+5 converge absolutely, and conditionally?Absolutely: k>−4Conditionally: −5<k≤−4
  17. ∑n=1∞(−1)n+1nConverges conditionally
  18. For what values of k does ∑n=1∞(−1)nnkn4 converge absolutely, and conditionally?Absolutely: k<3Conditionally: 3≤k<4
  19. ∑n=1∞(−1)n+1ln⁡(n+3)Converges conditionally
  20. For what values of k does ∑n=1∞(−1)nn2k−4 converge absolutely, and conditionally?Absolutely: k>52Conditionally: 2<k≤52

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

How to do these

∑n=1∞(−1)nn

  1. Without the signs, the terms are 1/n^(1/2): a p-series with p = 1/2, which diverges.
  2. With the signs, the terms alternate, shrink, and go to 0, so the alternating series test says it converges.
  3. It converges, but not absolutely.

The answer is Converges conditionally.

Where students go wrong

Stopping once the absolute values diverge. That rules out absolute convergence only; the alternating series can still converge conditionally.

Also called absolutely convergent series, conditionally convergent series or absolute convergence test.

What you can put on this worksheet

Absolutely, conditionally, or diverges
∑n=1∞(−1)n+1n+7 → Converges conditionally
The values of k for each kind of convergence
For what values of k does ∑n=1∞(−1)nnk+4 converge absolutely, and conditionally? → Absolutely: k>−3Conditionally: −4<k≤−3

Questions about these worksheets

Yes. Every absolute and conditional convergence 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 absolute and conditional convergence 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: absolutely, conditionally, or diverges and the values of k for each kind of convergence. Tick as many as you want and set how many of each, or let it spread them evenly.

Stopping once the absolute values diverge. That rules out absolute convergence only; the alternating series can still converge conditionally.