Power series and their radius of convergence

Take the limsup of the nth roots of the coefficient magnitudes. Multiplying a coefficient by the nth power of the distance from the center multiplies its nth-root size by that distance. The root test therefore proves absolute convergence when the resulting product is below one and divergence when it is above one. The reciprocal coefficient-growth rate is the radius, including the conventions that zero growth gives infinite radius and infinite growth gives zero radius.

The limsup handles missing coefficients and irregular growth. A ratio formula is convenient when coefficients are eventually nonzero and the consecutive ratio |cₙ₊₁/cₙ| has a limit, but it is not the definition and may fail to exist. For a series containing only even powers, ratios involving zero coefficients are unhelpful; the root formula still works directly. One should also distinguish coefficient index from the exponent when reindexing a sparse series, since an incorrect index changes the inferred radius.

Multiplying coefficients by a polynomial power of n does not change their nth-root growth, yet it can completely change convergence at the boundary. Coefficients one, one over n, and one over n squared all give radius one for expansions centered at zero. At x=1 their series are respectively divergent, divergent, and convergent. At x=−1, alternation changes the middle example to conditional convergence. A radius alone never specifies whether either real endpoint is included.

The same distinction matters for representing a named function. Summing a geometric series gives one over one minus x only inside its convergence interval. Algebraically extending that rational formula beyond the interval does not make the original series converge there. The function and a particular expansion are separate objects. Likewise, a smooth function need not equal its Taylor series: smoothness controls finite derivatives, while equality with an infinite Taylor expansion requires an appropriate remainder or analyticity argument.

Choose a compact interval whose distance from the center is at most r<R. A slightly larger radius still inside R yields geometric domination of the coefficient terms on that compact interval. The Weierstrass M-test then proves uniform convergence there. This is local uniform convergence: a single estimate works throughout every fixed compact subinterval, though it need not work on the whole open convergence interval. Near a boundary, the required number of terms can grow without bound.

Termwise differentiation and integration preserve the radius because multiplying or dividing coefficients by n changes their nth-root growth by a factor tending to one. Derivative identities are justified on compact interior intervals by uniform control, not by treating an infinite expression as an ordinary polynomial. In the spectral lab, changing coefficient decay similarly changes the contribution of high modes, but its sine basis is not a power basis and it does not compute a power-series radius.