the Cauchy-Hadamard formula
/ koh-SHEE ah-dah-MAR /
Given a power series, how do you actually find its radius of convergence? The Cauchy-Hadamard formula is the universal answer — a single explicit recipe that reads R straight off the coefficients, working for every power series without exception. It is the precise tool behind the loose intuition that 'fast-decaying coefficients reach far, fast-growing coefficients reach little'.
The formula is 1/R = limsup |a_n|^(1/n), with the conventions 1/0 = infinity and 1/infinity = 0. Here limsup (limit superior) is the largest value the sequence |a_n|^(1/n) keeps coming back to — you use it rather than an ordinary limit because the n-th roots of the coefficients may oscillate and need not settle to a single number, yet the limsup always exists. The derivation is exactly the root test applied to the series with z varying: the n-th term is a_n (z - z_0)^n, its n-th root of modulus is |a_n|^(1/n) |z - z_0|, and the root test demands this stay below 1, i.e. |z - z_0| < 1 / limsup |a_n|^(1/n). That threshold is R. When the simpler limit lim |a_(n+1)| / |a_n| exists, it gives the same R and is usually easier to compute, but Cauchy-Hadamard is the one that ALWAYS works.
Its importance is that it makes the radius of convergence a precise, computable, coefficient-only quantity — independent of any cleverness about the particular function. Combined with the fact that R equals the distance to the nearest singularity, Cauchy-Hadamard becomes a bridge: the analytic data (how fast coefficients grow) and the geometric data (where the function breaks) are two readings of the same number. It is named for Cauchy, who found it, and Hadamard, who emphasized and rediscovered it.
For sum 2^n z^n the coefficients are a_n = 2^n, so |a_n|^(1/n) = 2 for every n, the limsup is 2, and R = 1/2. The series sum (1 + (-1)^n) z^n has coefficients oscillating between 0 and 2; here |a_n|^(1/n) does not converge, but its limsup is 2^(1/n) -> ... settling to limsup = 1, giving R = 1 — a case where the ratio test fails but Cauchy-Hadamard still delivers.
When coefficients oscillate, the ordinary ratio limit may not exist, but the limsup of n-th roots always does — that is why the formula uses limsup.
You must use limsup, not an ordinary limit: the plain limit of |a_n|^(1/n) can fail to exist for perfectly nice series with gaps or oscillating coefficients, while the limsup is always defined and always gives the correct R.