Series Convergence Test Calculator

Test an infinite series

Choose the test that matches your series. Some tests require you to enter a limit that you have already calculated. The calculator applies the test to that value; it does not automatically derive limits from an arbitrary expression.

Select a convergence test

p-Series parameters

A standard p-series has the form Σ 1/n^p.

Exponent p Converges for p > 1 and diverges for p ≤ 1.

Geometric series parameters

For Σ a rⁿ, starting at n = 0, enter the first term and the common ratio.

First term a

Common ratio r

The sum is a/(1-r) when |r| < 1.

nth-term divergence test

Enter the known limit L = lim a_n.

Limit of the terms A nonzero limit proves divergence. A zero limit is inconclusive.

Ratio test

Enter the known limit L = lim |a_n+1/a_n|.

Ratio-test limit L Use a non-negative limit, including zero if appropriate.

Root test

Enter the known limit L = lim ⁿ√|a_n|.

Root-test limit L Use a non-negative limit.

Alternating series test

Consider a series whose signs alternate and whose non-negative term magnitudes are b_n.

Limit of b_n Enter the known limit of the non-negative magnitudes.

The magnitudes b_n are eventually non-increasing and non-negative. Check this only if the condition has been established mathematically.

Try a worked example

Load one of these examples to see how the calculator applies a convergence test.

Example 1: Convergent p-series

Σ 1/n²

Since p = 2 > 1, the series converges.

Example 2: Divergent harmonic series

Σ 1/n

Since p = 1, the series diverges.

Example 3: Geometric series

Σ (1/2)ⁿ, n = 0 to infinity

The ratio is 1/2 and the sum is 2.

Example 4: Ratio test

Σ 1/n!

The ratio limit is 0, so the series converges absolutely.

Example 5: Inconclusive ratio test

Σ 1/n²

The ratio limit is 1, so use another test.

Example 6: Alternating harmonic series

Σ (-1)ⁿ⁺¹/n

The magnitudes decrease to zero, so the series converges.

What is series convergence?

An infinite series is the sum of the terms of an infinite sequence. For a series Σ a_n, its partial sums are defined by S_N = a₁ + a₂ + ... + a_N.

The series converges if its partial sums approach a finite limit as N tends to infinity. Otherwise, it diverges. Convergence tests help establish this behavior without necessarily calculating the exact infinite sum.

Common convergence tests and formulas

Test	Convergence condition	What happens otherwise?
p-Series	p > 1	Diverges when p ≤ 1.
Geometric Series	\|r\| < 1	Diverges for nonzero first term when \|r\| ≥ 1.
nth-Term Test	Term limit must be zero for convergence.	A nonzero term limit proves divergence.
Ratio Test	L < 1 proves absolute convergence.	L > 1 proves divergence; L = 1 is inconclusive.
Root Test	L < 1 proves absolute convergence.	L > 1 proves divergence; L = 1 is inconclusive.
Alternating Series Test	Non-negative magnitudes eventually decrease to zero.	Failure of these conditions does not by itself prove divergence.

How to choose a convergence test

1. Check the terms first

Start by calculating lim a_n. If the limit is nonzero or does not exist, the series diverges. If it is zero, continue with another test.

2. Recognize familiar series

If the series is a geometric series, examine its common ratio. If its terms are 1/n^p, use the p-series test.

3. Use the ratio test for factorials

For series containing factorials or products of successive factors, the ratio a_n+1/a_n may simplify considerably. Take the limit of its absolute value.

4. Use the root test for powers

The root test is often useful when terms contain expressions raised to the nth power. Calculate the limit of the nth root of |a_n|.

5. Examine alternating series

For alternating terms, check whether their non-negative magnitudes decrease eventually and approach zero. If so, the alternating series test establishes convergence.

6. Remember the inconclusive cases

No single convergence test works for every series. In particular, a ratio or root limit equal to 1 requires a different method. Comparison, limit comparison, and integral tests can be useful alternatives when their conditions hold.

Frequently asked questions

Can a series converge if its terms approach zero?

Yes, but not always. Terms approaching zero are necessary for convergence, but the harmonic series Σ 1/n is a well-known example of a divergent series whose terms approach zero.

What does an inconclusive result mean?

It means the selected test does not establish convergence or divergence. It does not mean the series fails to converge. Another test may settle the question.

What is absolute convergence?

A series Σ a_n converges absolutely when Σ |a_n| converges. Absolute convergence implies convergence of the original series.

What is conditional convergence?

A series converges conditionally when it converges but the series of absolute values diverges. The alternating harmonic series is a standard example.

Does this calculator automatically analyze any formula?

No. This version applies selected convergence criteria to supplied parameters and known limits. It does not symbolically derive arbitrary limits or prove the hypotheses of a test automatically.

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