Compute the radius and open interval of convergence of a power series.
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Σ c_n x^n ， R = 1 / lim|c_{n+1}/c_n|
e.g. 1/n (n starts at 1)
Radius of Convergence
R =
📖 Tutorial | Power Series
1. Definition
A power series has the form Σ c_n x^n. By Abel's theorem there is a radius R so the series absolutely converges for |x|R. By the ratio test R=1/lim|c_{n+1}/c_n|.
2. Symbols
c_n
coefficients
R
radius of convergence
|x|
interval of absolute convergence
endpoints ±R
judged separately
3. How it works
Compute |c_{n+1}/c_n| at large n;
R is its reciprocal;
ρ=0 gives R=∞ (all reals), ρ→∞ gives R=0;
The interval is open; endpoints need separate tests.
4. Steps
Enter c_n, e.g. 1/n;
Click Calculate;
Read R and the open interval.
5. Example
Example: Σ x^n/n. Solution: |c_{n+1}/c_n|=n/(n+1)→1, so R=1, interval (-1,1). At x=−1 converges (alternating), at x=1 diverges (harmonic).
6. Pitfalls
The ratio test gives only the open interval; endpoints must be tested separately; With n! the radius is often ∞ (e^x); This expands about 0; shift x by x₀ for another center.
❓ FAQ | Power Series
How to find R?
R=1/lim|c_{n+1}/c_n|, or via the root test.
What does R=∞ mean?
Absolute convergence on the whole real line, e.g. Σx^n/n! = e^x.
Why judge endpoints separately?
At |x|=R the ratio test is inconclusive.
What if the center is not 0?
Replace x with (x−x₀); interval becomes |x−x₀|
Does this tool judge endpoints?
It reports the open interval; endpoints need extra tests.