Radius Of Convergence Calculator
Radius of Convergence Calculator
Slug: `/radius-of-convergence-calculator/`
Primary keyword: radius of convergence calculator
Meta title: Radius of Convergence Calculator: Ratio and Root Test, Full Steps
Meta description: Find the radius of convergence R for any power series. Ratio test and root test worked out step by step, with exact results where the form is recognised.
Word count: approximately 1,000
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What R actually measures
Every power series has a centre. Write it as a sum of a(n) times x minus c to the power n, and c is that centre. Plug x equal to c into the series and every term past the first disappears, so convergence at the centre is automatic.
Move away from c and the series keeps converging for a while. Then it stops. How far you get is the radius of convergence, R.
Think of it as a distance rather than a location. R equal to 3 for a series centred at 5 means convergence anywhere within 3 units of 5, so from 2 up to 8, endpoints pending.
Finding R with the ratio test
Apply the ratio test to the coefficients, not to the full terms. Strip off the x minus c part and work with a(n) alone.
Compute L as the limit of the absolute value of a(n+1) over a(n). Then R equals one divided by L.
Worked through on the series with coefficients one over n times two to the n:
Form the ratio of consecutive coefficients. The n over n plus one factor tends to 1, and the powers of two contribute a factor of one half. So L equals 0.5, giving R equal to 2.
Two special values need care. When L equals 0, R is infinite and the series converges everywhere. When L grows without bound, R equals 0 and the series converges at its centre alone.
Finding R with the root test
Take the nth root of the absolute coefficient, then take the limit. Again R equals one over that limit.
Root testing wins whenever the coefficient is itself an nth power. Coefficients of the form n over three, all raised to the power n, collapse immediately under a root: the nth root leaves n over three, which grows without bound, so R equals 0.
Attempt the same problem with the ratio test and you face a much uglier algebraic simplification for the same answer.
Three cases, five examples
| Series | Coefficient a(n) | L | R | Meaning |
|---|---|---|---|---|
| --- | --- | --- | --- | --- |
| sum x^n / n! | 1/n! | 0 | infinity | converges for all real x |
| sum x^n / n | 1/n | 1 | 1 | converges on (-1, 1) plus endpoint work |
| sum x^n / 2^n | 1/2^n | 0.5 | 2 | converges on (-2, 2) plus endpoint work |
| sum n^2 x^n | n^2 | 1 | 1 | converges on (-1, 1) plus endpoint work |
| sum n! x^n | n! | infinity | 0 | converges only at x = 0 |
Notice rows two and four. Different coefficients, same radius. Polynomial factors in a coefficient never change R, because any polynomial in n has ratio limit 1. That single observation saves a lot of algebra: you can discard polynomial factors when hunting for R and keep only the exponential and factorial parts.
Where numerical estimation struggles
Our calculator matches known coefficient forms symbolically before falling back on arithmetic, and there is a concrete reason for that preference.
Ratio limits converge slowly. Feed a numerical routine the coefficients one over n factorial and run 120 terms. The computed limit lands near 0.009. True value: exactly 0. Near enough to conclude the radius is infinite, nowhere near precise enough to present as a result.
Run the same routine on one over n and you get roughly 1.0092 instead of exactly 1, which would report R as 0.99 rather than 1. Fine for a sanity check. Not fine for homework.
So the engine labels every output. Exact means a recognised closed form. Estimated means arithmetic, and you should verify by hand before submitting it.
One line to remember
R is a distance. The interval is a set. They are not the same object.
What this tool cannot do
Our engine recognises geometric, factorial, p-series and polynomial coefficient forms exactly. Anything outside that list falls back to numerical estimation, and estimation cannot prove a radius. It also will not handle coefficients defined by a recurrence, piecewise rules, or complex values. For those, verify by hand or in a computer algebra system.
R is not the answer
Finding the radius completes half the problem. The interval of convergence needs both endpoints tested separately, because the ratio test returns L equal to 1 at each of them and therefore decides nothing.
For the series of x to the n over n, R equals 1, and the two endpoints behave differently. At x equal to 1 you get the harmonic series and divergence. At x equal to minus 1 you get the alternating harmonic series and convergence, to about minus 0.693147. Interval: from minus 1 inclusive to 1 exclusive.
Report R when the question asks for R. Report the interval, brackets and all, when the question asks for the interval.
Common errors
- Testing the full term instead of the coefficient. The x minus c factor belongs outside the ratio.
- Reporting R as an interval. R is a distance, a single non-negative number or infinity.
- Assuming R is finite. Exponential, sine and cosine series all have infinite radius.
- Forgetting the centre. R equal to 2 around c equal to 5 gives an interval near 5, not near 0.
- Keeping polynomial factors through the algebra. They cancel. Drop them early.
Questions
Can the radius be negative?
No. It is a distance, so it is zero, positive, or infinite.
Does the centre affect R?
No. Shifting c moves the interval along the line without changing its width.
Which test should I use?
Factorials favour the ratio test. Whole coefficients raised to the nth power favour the root test. Either works when both apply.
What if both tests give L equal to 1?
Then R equals 1, and the real work moves to the endpoints.
Why does the exponential series converge everywhere?
Factorial growth in the denominator outruns any fixed power in the numerator, driving the ratio limit to zero.