Cauchy–Hadamard theorem

How big is the radius of convergence? The Cauchy–Hadamard theorem answers this with a single, completely general formula. It looks at how fast the coefficients grow — measured by their n-th roots — and reads off the radius directly. No test-by-test guesswork: one formula, always correct.

The theorem states: for sum c_n (x - a)^n, let L = limsup_{n->infinity} |c_n|^{1/n}. Then the radius of convergence is R = 1/L, with the conventions R = infinity when L = 0 and R = 0 when L = infinity. The reason the limit superior (rather than an ordinary limit) appears is that |c_n|^{1/n} need not converge; limsup captures the largest persistent growth rate, which is exactly what controls the worst-case behavior of the terms.

This is more powerful than the ratio test, which uses limit of |c_{n+1}/c_n| and only works when that limit exists. The root-based limsup always exists in the extended reals, so Cauchy–Hadamard gives R for every power series with no exceptions. When the ratio limit does exist, it agrees with the root value, so the simpler ratio test is a valid shortcut whenever it applies.