Convergence Calculator

Determining whether an infinite series converges or diverges is essential in calculus, engineering, and physics, but manual testing often feels like guessing. The convergence calculator below eliminates trial and error: enter any series expression in terms of n, and it instantly identifies convergence or divergence using the most appropriate test.

Convergence Calculator Use n as variable: e.g., (-1)^n/sqrt(n), (2^n)/(n!), log(n)/n^2

The calculator accepts standard mathematical notation – factorials (n!), exponents (2^n), trigonometric functions, and summations like ∑ 1/n^2. It systematically applies up to 7 different convergence tests and returns a clear verdict together with the test that gave a decisive result. If the simplest tests fail, it moves to the limit comparison test or the integral test automatically. You simply type the general term and the lower index (default 1), then read the analysis.

How to use this convergence calculator

  1. Enter the general term – write the n‑th term of the series, for example 1/n^2, (2^n)/(n!), or (-1)^n / sqrt(n).
  2. Set the starting index – most series start at n = 1, but you can change it to any positive integer.
  3. Hit “Calculate” – the engine runs a sequence of convergence tests.
  4. Interpret the result – the calculator shows either “Converges” or “Diverges”, names the decisive test, and, for key steps, displays the limit used.

No sign‑up, no installation. The tool works on desktop and mobile browsers.

Which convergence test should I use for my series?

Choosing the right test manually is the main hurdle. The calculator does it for you, but understanding the logic helps you check its reasoning:

  • Ratio test – best when terms involve factorials or exponentials (e.g., n!/2^n). If the limit |aₙ₊₁ / aₙ| is < 1, the series converges; > 1, diverges.
  • Root test – ideal for terms raised to the nth power (e.g., (1+1/n)^(n^2)). Compute the nth root of the absolute term; limit < 1 signals convergence.
  • Limit comparison test – works when the series resembles a known p‑series or geometric series. Compare the limit of aₙ / bₙ with a known convergent/divergent bₙ.
  • Integral test – use when the term aₙ comes from a positive, continuous, decreasing function f(x) that you can integrate. The series converges if and only if ∫₁^∞ f(x) dx converges.
  • Alternating series test – automatically applies when there is a factor (-1)^n; the series converges if the absolute term decreases monotonically to zero.

For p‑series ∑ 1/n^p, the decision is instant: converge if p > 1, diverge if p ≤ 1.

Common convergence tests explained

The calculator incorporates seven classical tests, ranked by computational cost. Here is what each one checks:

1. Geometric series test

If the term can be written as a geometric progression a·r^n, the series converges when |r| < 1 and diverges otherwise. The calculator detects this structure immediately.

2. p‑series test

Any series of the form 1/n^p is tested directly: convergence for p > 1, divergence for p ≤ 1.

3. Ratio test

The limit L = limₙ→∞ |aₙ₊₁ / aₙ|. If L < 1, converges; L > 1, diverges; L = 1, inconclusive.

4. Root test

L = limₙ→∞ |aₙ|^(1/n). Same interpretation as the ratio test.

5. Integral test

If aₙ = f(n) with f(x) continuous, positive, and decreasing, then convergence of ∑ aₙ is equivalent to convergence of ∫₁^∞ f(x) dx. The calculator approximates the integral numerically when necessary.

6. Limit comparison test

Given a known series bₙ (often a p‑series), compute lim (aₙ / bₙ). If the limit is finite and non‑zero, both series converge or diverge together.

7. Alternating series test

For series with (-1)^n factor, check that |aₙ| → 0 and decreases eventually. If yes, series converges.

The engine applies these in order and stops at the first conclusive result.

Example: testing ∑ 1/n² for convergence

Enter 1/n^2 with n from 1 to infinity. The calculator first identifies the p‑series pattern with p = 2. Since 2 > 1, it immediately reports Converges. It also gives a quick estimate of the sum: approximately 1.6449 (which is π²/6). If you change the exponent to 1, the same series becomes the harmonic series – the tool returns Diverges because p = 1 ≤ 1.

For a more challenging case, try 2^n / n!. The ratio test gives:

|aₙ₊₁ / aₙ| = (2^(n+1)/(n+1)!) / (2^n/n!) = 2/(n+1) → 0 < 1

so the calculator concludes convergence and may note that the series sums to e² - 1.

Why choose an online convergence calculator?

Manual testing is error‑prone and time‑consuming. Students often misapply the ratio test when a simpler p‑series check would suffice. Professionals verifying series convergence for signal processing or financial models need quick, reliable results. This calculator gives you both speed and pedagogical clarity – it shows which test worked, helping you learn while you work.

The calculator provides mathematically sound results based on standard convergence tests. Always verify outcomes for new or unusual series through further analysis if you rely on the conclusion for critical applications.

Frequently Asked Questions

Can this calculator test a power series like ∑ xⁿ/n!?
Yes, you can enter power series by using the variable n inside the term expression. The calculator treats other symbols as constants. For series with parameters, it may return conditional convergence if the test depends on the parameter’s value.
What does “inconclusive” mean in the result?
Some tests, like the ratio or root test, yield a limit of 1, which gives no information about convergence. The calculator then tries alternative tests. If all applicable tests are inconclusive, it reports that outcome and suggests more advanced methods.
Does the convergence calculator handle alternating series?
Absolutely. The calculator detects alternating signs (e.g., (-1)^n) and automatically applies the alternating series test. It checks if the absolute value of terms decreases to zero, which guarantees convergence.
How accurate is the result?
The calculator uses analytically proven convergence tests, so its conclusions are mathematically reliable for standard real-valued series. However, for highly irregular or conditionally convergent series, you should verify using additional analysis.
Can I input factorials and exponentials?
Yes. The expression parser supports factorial notation (n!), exponentials (a^n or exp(n)), logarithms, trigonometric functions, and parentheses. Enter the general term as a function of n, and the calculator will evaluate it for convergence testing.
Is convergence of an improper integral the same as series convergence?
Not always, but the integral test links them: if a continuous, positive, decreasing function f(x) gives a series ∑ f(n), the series converges only when the improper ∫₁^∞ f(x) dx converges. The calculator applies this test when the term structure allows integration.
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