Convergence Calculator
Ratio, root, p-series, geometric, alternating

Does your series converge?

Enter a series and get the verdict, the test that decided it, the radius and interval, and the endpoint check that most calculators quietly skip.

Free, no sign-up Runs in your browser Exact answers labelled exact Every step shown
Convergence engine ready
Term a(n) ?
Index range

The engine recognises geometric, p-series, factorial and polynomial forms in closed form. Anything else falls back to a numerical estimate, and it says so.

Verdict
Ready

Enter a term to begin.

Partial sums approaching the limit

Chart of partial sums against the number of terms

Numerical results are estimates and are labelled as such. Exact results come from recognised closed forms. Verify before submitting as proof.

What a convergence calculator actually decides

A convergence calculator determines whether an infinite series adds up to a finite number or grows without bound. It applies standard tests, the ratio test, root test, p-series test, comparison test and alternating series test, and for a power series it returns the radius of convergence R together with the interval of x values where the series behaves. Convergence guarantees a finite sum exists. It does not promise that sum has a formula you can write down.

The toolkit

Six tests, and the shape each one is built for

Picking the wrong test is the fastest way to lose marks. Match the structure of the term to the test designed for it.

TEST 01

Ratio test

Take the limit of the absolute ratio of consecutive terms.

L = lim |a(n+1) / a(n)|

Use when the term contains a factorial or something raised to the power n. Silent when L equals 1.

TEST 02

Root test

Take the nth root of the absolute term, then the limit.

L = lim |a(n)|^(1/n)

Use when the whole term is raised to the nth power. Handles cases the ratio test fumbles.

TEST 03

p-series test

The cleanest verdict in the entire subject.

sum 1/n^p converges iff p > 1

Use when the term is a power of n. At p equal to 1 you have the harmonic series, which diverges.

TEST 04

Geometric series

The rare case that hands you an exact sum.

sum ar^n = a/(1-r) when |r| < 1

Use when each term is a fixed multiple of the one before it.

TEST 05

Alternating series

Signs flip, terms shrink to zero, the series settles.

b(n) decreasing and b(n) to 0

Use when the term carries a factor of minus one to the n. Gives conditional convergence.

TEST 06

Divergence test

The ten second check to run before anything else.

If a(n) does not tend to 0, it diverges

Use when starting any problem. It can rule convergence out, never confirm it.

Where the marks go

The endpoint check

Finding R is the easy half. Your interval is decided at two points the ratio test cannot touch.

Run the ratio test on a power series and you get a radius R. That gives the open interval from c minus R to c plus R, where the series converges absolutely. Students frequently stop there, write an open interval, and lose the mark.

At each endpoint the ratio limit equals exactly 1, and the ratio test says nothing when L equals 1. Both endpoints have to be substituted back into the original series and tested by a different method. Four outcomes are possible, and all four occur in practice.

Endpoint outcomes for power series with radius 1
SeriesRLeft endRight endInterval
sum x^n1divergesdiverges(-1, 1)
sum x^n / n1convergesdiverges[-1, 1)
sum x^n / n^21convergesconverges[-1, 1]
sum (-1)^n x^n / n1divergesconverges(-1, 1]

Study the second row. Radius 1, and the two endpoints disagree. Substituting x equal to 1 gives the harmonic series, which diverges. Substituting x equal to minus 1 gives the alternating harmonic series, which converges to negative log 2, about minus 0.693147. Identical radius, opposite verdicts, and the interval is half open because of it.

The radius is arithmetic. The interval is judgement.

Reading the result

Three things worth knowing

Convergence is not the same as having a sum

Sum one over n squared and you converge to pi squared over six, roughly 1.644934, a result Euler proved in 1735. Sum of one over n cubed also converges, to about 1.202057, and no elementary expression for it has ever been found. Both converge. Only one has a formula.

Absolute against conditional

When the series of absolute values converges, the original converges absolutely, and its terms can be rearranged freely without changing the sum. When the original converges but the absolute version does not, convergence is conditional. Riemann showed that a conditionally convergent series can be reordered to sum to any number you choose. The alternating harmonic series adds to about 0.693147 in its natural order, and its terms can be shuffled to reach 5, or minus 12, or pi.

Why numerical evidence misleads

Add two hundred thousand terms of the harmonic series and you reach about 12.78. That looks like flattening. It is not. Partial sums grow like the natural log of n, so passing 100 would take roughly ten to the power 43 terms. Our engine prefers a symbolic match precisely because arithmetic alone cannot settle borderline cases.

Answers

Questions people actually ask

What is a convergence calculator?
A convergence calculator decides whether an infinite series adds up to a finite number. It applies standard tests such as the ratio test, root test, p-series test and alternating series test, and for a power series it finds the radius and interval of convergence.
Why is the ratio test inconclusive at the endpoints?
At an endpoint the ratio limit L equals exactly 1, and the ratio test says nothing when L equals 1. That is why every endpoint has to be substituted back into the original series and tested with a different method.
What is the difference between radius and interval of convergence?
The radius R is a single number giving the distance from the centre over which the series converges. The interval is the actual set of x values, which requires checking whether each endpoint is included or excluded.
Does a convergent series always have a sum you can write down?
No. Convergence guarantees a finite sum exists, not that it has a closed form. Geometric and telescoping series have simple sums. Most others, including the sum of one over n cubed, converge to constants with no elementary expression.
Which test should I try first?
Run the divergence test, since it costs nothing. If the terms do not tend to zero you are finished. After that, match the shape: factorials or powers of n point to the ratio test, an entire term raised to the nth power points to the root test, a plain power of n is a p-series, and a factor of minus one to the n means the alternating series test.
Can a series diverge if its terms tend to zero?
Yes. The harmonic series has terms tending to zero and still diverges, which is exactly why the divergence test can only rule convergence out rather than confirm it.
What does a radius of zero mean?
The series converges only at its centre and nowhere else. The standard example is the sum of n factorial times x to the n, which converges at x equal to 0 and diverges for every other value.
How accurate is the numerical fallback?
Treat it as a strong hint rather than proof. Ratio limits converge slowly: the coefficients of the series for e to the x give a numerical L near 0.009 after 120 terms when the true limit is 0. Where the engine matches a known closed form it reports exact values, and it labels every estimate.

Go further

Related calculators

Radius of ConvergenceFind R by ratio or root test Interval of ConvergenceR plus both endpoints Ratio TestLimit of consecutive terms Root TestFor nth power terms p-Series TestConverges when p exceeds 1 Geometric SeriesExact sum when the ratio is small Alternating SeriesConditional convergence Taylor SeriesExpand and find where it holds