Series Convergence Calculator

Choose a valid convergence test, check its conditions, and classify an infinite series.

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Enter a series, such as sum n/(n^3 + 1) from n=1 to infinity

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A convergence verdict needs a proof

An infinite series converges when its sequence of partial sums approaches a finite value. Terms approaching zero are necessary, but that condition alone is not enough, as the harmonic series shows.

Different structures call for different tests. Geometric and p-series patterns can settle a problem quickly. Comparison, integral, ratio, root, and alternating-series tests each have conditions that must be checked before their conclusion is valid.

How to use the series convergence calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

The nth-term divergence test

If the terms do not approach zero, the series diverges. A zero term limit is only a first check and does not prove convergence.

n \to \infty lim a_{n} \neq = 0 ⟹ n = 1 \sum \infty a_{n} diverges

A reliable way to work through it

Inspect the term pattern

Look for geometric, p-series, alternating, factorial, exponential, or rational behavior before selecting a test.

Verify the test conditions

Show positivity, monotonicity, comparison bounds, or the required limit instead of naming a test without its hypotheses.

Classify the conclusion

Distinguish absolute convergence, conditional convergence, divergence, and an inconclusive test.

Worked example

Test Σ n/(n³ + 1) from n = 1 to infinity

b_{n} = \frac{1}{n ^{2}}

Compare with a convergent p-series.

lim_{n \to \infty} \frac{n / ( n ^{3} + 1 )}{1/ n ^{2}} = lim_{n \to \infty} \frac{n ^{3}}{n ^{3} + 1} = 1

Compute the limit comparison ratio.

\sum_{n = 1}^{\infty} \frac{1}{n ^{2}} converges

The comparison series has p = 2, which is greater than 1.

The series converges by the limit comparison test. Its terms are positive, so the convergence is absolute.

Common mistakes to check

Using a zero term limit as proof

The condition aₙ → 0 is necessary but not sufficient for series convergence.

Treating a test value of 1 as a verdict

A ratio-test or root-test result of 1 is inconclusive and requires another method.

Skipping endpoint tests

A power-series radius gives an open interval. Each boundary point must be tested separately.

Limit CalculatorStudy what a function approaches from the left, right, or at infinity, even when direct substitution fails.Taylor Series CalculatorBuild a Taylor or Maclaurin polynomial from a function, expansion center, and chosen degree.Definite Integral CalculatorEvaluate an integral between two bounds and separate signed accumulation from geometric area.Integral CalculatorFind an indefinite integral, follow the chosen method, and keep the constant of integration.

Questions students ask

Does aₙ approaching zero prove convergence?

No. It is necessary but not sufficient. The harmonic series is the standard counterexample.

What is absolute convergence?

A series converges absolutely when the series of absolute values converges. Absolute convergence implies ordinary convergence.

What if the ratio or root test gives 1?

That test is inconclusive. A comparison, integral, alternating, or another suitable test may still decide the series.

Does convergence mean the exact sum is known?

No. A series can be proven convergent without having a simple closed-form sum.