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Inside its radius of convergence, a power series $f(z)={\sum }_{n\ge 0}{a}_n(z-c{)}^n$ can be differentiated term by term:The differentiated series has the same radius of convergence. The conclusion concerns the interior, including uniform convergence on smaller closed disks; it does not assert that differentiated boundary series converge wherever the original did. For example, ${\sum }_{n\ge 1}{z}^n/n$ converges conditionally at $z=-1$, but its differentiated series ${\sum }_{n\ge 1}{z}^{n-1}$ does not.

Applied to the geometric series, it gives the useful identityDifferentiation turns the plain coefficients into integer weights. This is one way generating functions translate a weighted counting question into an operation on a known series. A proof controls the original and differentiated tails on smaller disks; the Weierstrass M-test provides the needed uniform bounds. See Lebl, Basic Analysis, §11.3: power-series differentiation and analyticity.

The Cauchy product has coefficients $c_n={\sum }_{k=0}^n{a}_k{b}_{n-k}$. Absolute convergence permits regrouping and makes its sum the product of the two sums.

The radius of convergence of ${\sum }_{n\ge 0}{a}_n(z-c{)}^n$ is the unique $R\in [0,\infty ]$ separating interior absolute convergence from exterior divergence. It converges absolutely when $|z-c|<R$ and diverges when $|z-c|>R$. At the centre it always has the finite value $a_0$, since all positive powers vanish. The boundary convergence of a power series is a separate question.

For $0<R<\infty$, the interior is a disk in the complex plane and an open interval $(c-R,c+R)$ on the real line. A power series converges uniformly on every smaller closed disk $|z-c|\le r<R$. This gives stable polynomial approximation over a whole compact region, rather than a separate limiting calculation at each point. See Jiří Lebl, Basic Analysis, §11.3.2: convergence of power series.

Four examples reveal what $R$ measures:

The Cauchy-Hadamard theorem computes $R$ directly from the coefficients. The attractive feature is that one growth statistic decides convergence throughout the interior and exterior, including sparse series with many zero coefficients. Finite changes to the coefficients do not alter the radius, though they can change the sum.

Subspace iteration repeatedly applies a matrix to several independent vectors and orthonormalizes them, converging to a dominant invariant subspace when a spectral gap separates the selected eigenvalue cluster.

Inverse iteration repeatedly solves $(A-sI)y_{k+1}=q_k$ and normalizes. It is the power method for $(A-sI{)}^{-1}$ and converges toward an eigenvector whose eigenvalue is closest to the shift $s$.

For a diagonalizable matrix, the active spectrum consists of eigenvalues whose eigenspaces have nonzero components of the starting vector in its spectral decomposition. These components are orthogonal projections for a self-adjoint operator, but need not be orthogonal projections for a general diagonalizable matrix. With a unique largest modulus in this active spectrum, the power method selects its eigendirection; an exactly absent dominant component is never created in exact arithmetic.

For a real symmetric matrix with $|{\lambda }_1|>|{\lambda }_2|$ and a starting vector having nonzero component in the leading eigendirection, power-method direction errors arewhereas the Rayleigh quotient error isThe improvement occurs because orthogonality cancels terms linear in the direction error.

Shallow-water potential vorticity is materially conserved by inviscid, unforced shallow water equations. Linearizing about rest with depth $H_0$ gives an anomaly proportional to

The second potential slow-roll parameter isSlow roll usually requires both ${ϵ}_V\ll 1$ and $|{\eta }_V|\ll 1$.

If $A=Q\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)Q^T$ is real symmetric and positive semidefinite, its principal square root isIt is symmetric and positive semidefinite, and is positive definite exactly when $A$ is.

When $b_j=1$ and $a_j$ are positive integers, adjacent convergents satisfyEven and odd convergents are monotone from opposite sides, their separation is $1/(q_n{q}_{n+1})$, and $q_n\to \infty$. They therefore have a common limit $x$, with

A complex matrix $A$ is Hermitian positive definite when $A^{\ast }=A$ and $z^{\ast }Az>0$ for every nonzero complex vector $z$.

For a real symmetric positive-definite matrix,It measures the sensitivity of the linear system and controls convergence bounds for iterative solvers.

A Hermitian matrix is positive definite exactly when all its leading principal minors are positive.

Forset $a=U(t)b$ withThen $b$ evolves under the constant Hamiltonian

The Barenblatt solution is a mass-preserving similarity solution of the porous medium equation. For $u_t=\beta (u{u}_x{)}_x$ in one dimension it has the form

To approximate a piecewise constant function pointwise, exhaust each of its separated regions by compact subsets whose finite union has connected complement. Apply polynomial Runge approximation to the locally constant holomorphic function on each stage, with errors tending to zero. Every fixed point eventually lies in all later stages, so the uniform stagewise estimates imply pointwise convergence.

A proper closed arc $S$ of a circle centered at zero has connected complement and stays away from zero. The polynomial Runge theorem therefore gives polynomials converging uniformly to $1/z$ on $S$.

A regular hyperbolic octagon with opposite sides paired gives a genus-two surface when each angle is $\pi /4$. The eight vertices then form one smooth quotient point because their angles sum to $2\pi$.

Identifying the $2n$ sides of one polygon in pairs gives a cell structure with one face, $n$ edges and some number $V$ of vertex classes. For a closed orientable surface of genus $g$,Since $V\ge 1$, every such schema satisfies $n\ge 2g$.