Summation Convergence Calculator

Enter Summation Expression (use n as the variable):

Lower Limit (n =):

Calculate

Result:

In mathematical analysis, especially in calculus and infinite series, determining whether a summation (or series) converges or diverges is fundamental. A convergent series approaches a finite number, while a divergent series either grows without bound or oscillates indefinitely.

This Summation Convergence Calculator helps you check whether a series like:

Σ (1/n²) or Σ (1/ln(n)) converges or diverges. It evaluates the summation from a starting index and adds terms until either convergence is clear or divergence is detected.

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Formula

The general form of an infinite series is:

Σ from n = a to ∞ of f(n)

• If the partial sums approach a finite value as n → ∞, the series converges.
• If the sum grows without bound or does not settle, the series diverges.

There is no one-size-fits-all formula. Instead, convergence is often determined by:

• Comparing to known convergent/divergent series
• Applying convergence tests (like ratio test, p-series test, integral test)
• Numerical simulation (what this calculator does)

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How to Use

1. Enter the expression using n as the variable (e.g., 1/n^2, 1/(n*log(n)))
2. Set the lower limit (default is 1)
3. Click "Calculate"
4. The calculator will show:
   • Converges ≈ value (if the sum stabilizes)
   • Diverges (if it increases or behaves wildly)

⚠️ Use n only, not x or other variables. Use valid JavaScript math syntax:

• ^ is supported via JS exponentiation: use Math.pow(n, 2) or n**2
• Use Math.log(n) for natural logarithm

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Example

Example 1: Convergent Series

Input: 1/n^2
Lower Limit: 1
Result: Converges ≈ 1.644934
(This is the Basel problem, which converges to π²/6)

Example 2: Divergent Series

Input: 1/n
Lower Limit: 1
Result: Diverges
(The harmonic series diverges slowly)

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FAQs

1. What is a convergent series?
A series where the infinite sum approaches a finite number.

2. What is a divergent series?
A series whose sum keeps growing or oscillating and doesn’t settle.

3. What format should I enter the expression in?
Use valid JavaScript-style math with n as the variable.

4. What if I use an invalid expression?
You’ll get an alert. Make sure your syntax is correct.

5. What is the range of terms this calculator uses?
It sums up to a million terms or until the term is smaller than 1e-10.

6. Can I enter log or ln?
Yes, use Math.log(n) for natural log.

7. What if I use 1/n^0.5?
This diverges and the calculator will return Diverges.

8. Does this use exact convergence tests?
No, it simulates the summation numerically for fast estimation.

9. Can I use factorials?
Yes, but you'd need to implement them in expression form, like 1/(n*(n+1)).

10. Can I use this for alternating series?
Yes! Try ((-1)^n)/n for the alternating harmonic series.

11. Can I use Math.sin(n) or trig functions?
Yes, but they often cause oscillations leading to divergence.

12. Will it detect oscillating divergent series?
Yes, usually if the values don’t settle, it returns Diverges.

13. Can I see the partial sums?
Not in this version. This only shows the final result.

14. What’s the stopping criteria?
If terms drop below 1e-10, it assumes convergence.

15. What if I use negative starting index?
It may compute incorrectly. Use positive lower limits.

16. Does it support exponential growth?
Yes. Try 1/(2^n) for a convergent geometric series.

17. Why does 1/n diverge while 1/n^2 converges?
Because 1/n is a harmonic series (p = 1), and convergence requires p > 1.

18. Can I calculate finite sums?
This tool is focused on infinite series. For finite sums, just use a different calculator.

19. Is this calculator accurate?
It gives highly accurate estimates for most common series.

20. Is this tool free to use?
Yes, completely free and browser-based.

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Conclusion

The Summation Convergence Calculator is a fast and simple tool for testing whether your series converges or diverges. Whether you're studying calculus, exploring number theory, or just curious, this tool provides immediate insight into infinite summations.