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Ratio & Root Tests Calculator

Compute the ratio and root limits for a positive-term series to decide convergence.

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ρ=|a_{n+1}/a_n| ， r=ⁿ√|a_n|
e.g. 2^n/n!

📖 Tutorial | Ratio & Root Tests

1. Theorem
Ratio test: ρ=lim|a_{n+1}/a_n|. ρ<1 converges, ρ>1 diverges, ρ=1 inconclusive. Root test: r=lim ⁿ√|a_n|. r<1 converges, r>1 diverges, r=1 inconclusive.
2. Symbols
ρratio limit |a_{n+1}/a_n|
rroot limit ⁿ√|a_n|
<1converges
>1diverges
3. How it works
  • Approach the limit at n=20000;
  • Compute |a_{n+1}/a_n| = ρ;
  • Compute |a_n|^(1/n) = r;
  • Both are judged against 1.
4. Steps
  1. Enter the term a_n;
  2. e.g. 2^n/n!;
  3. Click Calculate;
  4. Compare ρ and r with 1.
5. Example
Example: Σ 2^n/n!.
Solution: ρ=2/(n+1)→0<1, so the series converges; the root r→0 also gives convergence.
6. Pitfalls
When ρ=1 or r=1 the tests fail—use comparison or integral test;
Ratio test is best with factorials and exponentials;
If ρ>1 the term does not tend to 0, so it diverges.

❓ FAQ | Ratio & Root Tests

Which test to prefer?
Ratio for n! and a^n; root when the term is an n-th power.
What if ρ=1?
Both fail, e.g. p-series; use integral or comparison test.
Is n=20000 accurate?
Enough for most problems; if ρ is extremely close to 1, use a formal test.
Why does ρ>1 force divergence?
Then |a_{n+1}|>|a_n|, so the term does not tend to 0.
Can I use it on alternating series?
Apply it to |a_n|; ρ<1 means absolute convergence.