Telescoping Series Calculator
Enter a telescoping series of the form 1/(n*(n+k)) to see partial fractions, term-by-term cancellation, and the exact partial and infinite sums.
Use the Telescoping Series Calculator
Series Parameters
Series
∑ 1 / (n × (n + 1))
Series: ∑ 1/(n×(n+k))
Between 2 and 20
Partial Sum S(N)
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Decimal
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Infinite Sum S(∞)
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Partial Fraction Decomposition
Term-by-Term Cancellation
| n | Term 1/(n(n+k)) | Partial fraction | Survives? |
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Collapsed Sum
Summary
Enter a telescoping series of the form 1/(n*(n+k)) to see partial fractions, term-by-term cancellation, and the exact partial and infinite sums.
How it works
- Enter the gap k (the spacing between factors) for the series sum 1/(n*(n+k)).
- Choose a positive starting index and the number of terms to display.
- The tool applies partial fraction decomposition: 1/(n*(n+k)) = (1/k)*(1/n - 1/(n+k)).
- Terms are expanded row by row so you can watch intermediate fractions cancel.
- The partial sum and S(∞) = (1/k) × sum from m=start through start+k−1 of 1/m are displayed.
Use cases
- Check homework answers for telescoping series problems.
- Visualize why telescoping series converge to a clean closed form.
- Verify partial fraction decomposition by expanding the series.
- Explore how changing k shifts the cancellation pattern.
- Use as a teaching aid to demonstrate series collapsing.
- Quickly compute partial sums without manual arithmetic.
Frequently Asked Questions
Last updated: 2026-09-26 ·
Reviewed by Nham Vu