Root Test

The Root Test

The root test (also called Cauchy's root test) determines whether a series $\sum a_n$ converges by computing:

$L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} |a_n|^{1/n}$

Conclusion:

• $L < 1$ → series converges absolutely
• $L > 1$ → series diverges
• $L = 1$ → inconclusive (try another test)

Comparison with the Ratio Test

Both tests use the same threshold ($L < 1$ converges, $L > 1$ diverges), but the root test is more powerful — it gives a conclusion in some cases where the ratio test is inconclusive. It is especially useful when $a_n$ is a power or exponential expression raised to the $n$-th power.

Examples

$\sum (1/2)^n$ — geometric series: $L = \lim_{n\to\infty} \left(\frac{1}{2^n}\right)^{1/n} = \frac{1}{2} = 0.5 < 1 \implies \text{converges}$

$\sum 2^n$ — divergent geometric: $L = \lim_{n\to\infty} (2^n)^{1/n} = 2 > 1 \implies \text{diverges}$

$\sum n^{-n}$ — ultra-fast convergence: $L = \lim_{n\to\infty} (n^{-n})^{1/n} = \lim 1/n = 0 < 1 \implies \text{converges}$

Numerical Approximation

For large $n$, $|a_n|^{1/n}$ is a good approximation of $L$:

double root_test_limit(double (*a)(int), int n) {
    return pow(fabs(a(n)), 1.0 / (double)n);
}

Your Task

Implement double root_test_limit(a, n) that computes $|a_n|^{1/n}$ for a given term function $a$ and large $n$.