Linked from

The 21 pages that link to Radius of convergence, each with the reason it gives.

• Taylor seriesRelated: It specifies how far from the expansion point the Taylor series can converge.

• Power seriesRelated: It marks the interval or disk where the series defines a function.

• Analytic continuationRelated: A Taylor series may stop converging before the function itself stops being continuable.

• Taylor's theoremRelated: It governs where an infinite Taylor series converges, a question separate from the finite theorem.

• Ratio testRelated: Applying the ratio test to coefficients often computes this radius.

• Multipole expansionRelated: The nearest source or singularity limits where a multipole representation converges.

• Probability-generating functionRelated: The function's analytic domain determines where evaluations and derivative arguments are valid.

• Root testRelated: Applying the root test to power-series terms yields the radius of convergence.

• Taylor polynomialRelated: For an analytic function, it helps determine where its Taylor series represents the function.

• Abel's theorem (power series)Related: Convergence at the boundary point 1 is the extra hypothesis beyond interior convergence.

• Cauchy–Hadamard theoremRelated: The theorem computes this radius directly from the coefficients.

• Binomial seriesRelated: For noninteger exponents, the binomial series converges for \(|x|<1\), with endpoints requiring separate tests.

• Lagrange inversion theoremRelated: Formal coefficient identities do not by themselves determine where the resulting series converges.

• Abel's testRelated: Abel's theorem helps describe behavior at the boundary not determined by the radius alone.

• Virial expansionRelated: It limits how far in density the virial series can be trusted.

• Hadamard gap theoremRelated: The theorem identifies this radius's boundary as a barrier to analytic continuation.

• Analytic functionRelated: Each local power-series representation is valid only within its convergence region.

• Analyticity of holomorphic functionsRelated: For a holomorphic function, the nearest singularity limits how far its Taylor expansion can extend.