For $p,q>0$, the positive series converges precisely when $p>1$, or $p=1$ and $q>1$. For $p>1$ use comparison with a convergent p-series; for $0<p<1$, the bound $(\log n{)}^q<n^{(1-p)/2}$ gives a lower comparison with a divergent p-series. At $p=1$, the integral test for convergence and $u=\log x$ reduce the question to $\int u^{-q}du$. In contrast, ${\sum }_{n\ge 3}[n(\log \log n{)}^r{]}^{-1}$ diverges for every finite $r>0$, since $(\log \log n{)}^r<\log n$ eventually and $\sum [n\log n{]}^{-1}$ diverges.

A factorial series with digits $0\le a_n<n$ is an absolutely convergent series of real numbers between zero and one. The telescoping series identityshows that its largest possible tail after place $N$ is $1/N!$. Multiplying this tail by $N!$ gives a number between zero and one, a useful arithmetic constraint on rational numbers.

A rule for assigning a value to some series through a specified limiting operation. It must say which series belong to its domain; different rules can have different domains. A regular summation method agrees with ordinary summation for every convergent series.

A series whose partial sums do not have a finite limit of a sequence. They may grow without bound or oscillate. Some divergent series belong to the domain of a summation method, such as Cesàro summation.

A binary sequence is a finite or infinite sequence whose terms belong to $\{0,1\}$.

Two bounded real sequences with the same set of subsequential limits admit permutations that match their terms with differences tending to zero. Alternate between taking the least unused index of the first sequence and of the second. Match its value to an unused term of the other sequence near a nearest common cluster point, with additional error at most $1/j$ at stage $j$. Every cluster-point neighbourhood contains infinitely many available terms. The actively chosen indices tend to infinity, whose distance to the common cluster set tends to zero by boundedness. By step $2j$, both lists have included every index up to $j$.

Here $s_2(n)$ is the sum of the binary digits of the nonnegative integer $n$. The sign version is $f(n)=(-1{)}^{s_2(n)}$. Concatenating binary digits gives $f(a{2}^j+r)=f(a)f(r)$ for $0\le r<2^j$. It supplies cancellation controlled by binary blocks rather than multiplicative factorization.

A recurrence relation specifies members of a sequence in terms of previous members, together with enough initial values to determine the sequence. A three-term recurrence for orthogonal polynomials relates consecutive orders. For example, the physicists' Hermite polynomials satisfy $H_{n+1}(x)=2x{H}_n(x)-2n{H}_{n-1}(x)$, starting from $H_0=1$ and $H_1=2x$.

The limit inferior and limit superior retain the lower and upper long-run behaviour of a real sequence, even when it has no ordinary limit of a sequence. For each tail of a sequence beginning at $N$, defineDiscarding more initial terms raises or preserves the infimum and lowers or preserves the supremum. Hence $(l_N)$ is nondecreasing and $(u_N)$ is nonincreasing. Their limits exist in the extended real number line:For a bounded sequence, both values are finite and $\mathrm{liminf}{a}_n\le \mathrm{limsup}{a}_n$. These bounds use the entire infinite tail of a sequence, rather than the last few observed terms. ^[ref].

Take the original example $a_n=(-1{)}^n(1+1/n)$, beginning $-2,3/2,-4/3,5/4,\dots$. The subsequence at even integers decreases toward one, while the subsequence at odd integers increases toward minus one. If $e_N$ is the first even integer at least $N$ and $o_N$ is the first odd integer at least $N$, thenThus $\mathrm{liminf}{a}_n=-1$ and $\mathrm{limsup}{a}_n=1$. The finite initial extreme $a_1=-2$ disappears from later lower bounds; neither long-run value is attained by any term of this example.

For a bounded sequence, the limit inferior is the smallest subsequential limit and the limit superior is the largest subsequential limit; both are reached by suitable convergent subsequences. Every other subsequential limit lies between them. In the example there are only two subsequential limits, $-1$ and $1$: a convergent subsequence cannot retain infinitely many terms of both parities, because those two branches stay separated. The whole interval between the two bounds need not consist of subsequential limits. ^[ref].

A finite ordinary limit of a sequence exists exactly when the limit inferior and limit superior agree at the same finite value $L$. The bounds $l_N\le a_N\le u_N$ and the squeeze theorem prove the sufficient direction. For a bounded sequence, its tail diameter of a sequence is $D_N=u_N-l_N$, so the limiting gap is zero exactly when the Cauchy sequence condition holds. In the example the gap approaches two, not zero. The same framework allows infinite bounds: $a_n=n$ has both values $+\infty$, whereas $a_n=(-1{)}^{n}n$ has limit inferior $-\infty$ and limit superior $+\infty$. Equality at infinity describes escape rather than convergence to a real value. ^[ref].

A tail of a sequence consists of its terms from some index $N$ onward, retaining their order and repetitions. Discarding finitely many initial terms does not change its limit of a sequence or whether it is a Cauchy sequence. A tail can also be considered as the set of values $\{x_n:n\ge N\}$ when forming an infimum, supremum, or tail diameter of a sequence; that set alone forgets the original order and repetitions.

A sequence indexed from zero with an integer period $p\ge 1$ satisfying the displayed identity for all $n\ge 0$. Repeating a finite cycle is an example. The Cesàro mean article explains why its cumulative averages approach the one-cycle arithmetic mean.

A constant sequence has the same value $a$ at every index. It converges to $a$ by convergence of a sequence in a topological space.

An algebra homomorphism onto a semisimple Banach algebra is continuous. To see the spectral mechanism, suppose $a_n\to 0$ and $T(a_n)\to b$. For any $c\in B$, choose $u,v\in A$ with $T(u)=b$ and $T(v)=c$, and set $d=cb$, $d_n=T(v{a}_n)$. Apply the spectral-radius three-circle inequality to $T((1-z)v{a}_n+zvu)$. On the outer circle its norm tends uniformly to $\Vert d\Vert$; on the circle of radius $1/R$ its spectral radius is bounded by $(1+1/R)\Vert v{a}_n\Vert +\Vert vu\Vert /R$. Hence $r(d{)}^2\le \Vert d\Vert \Vert vu\Vert /R$ for every $R>1$, so $r(cb)=0$. The unit criterion for the Jacobson radical gives $b\in J(B)=0$, and the closed graph theorem finishes the proof.

For a uniform piecewise-linear Galerkin heat discretization, the consistent mass matrix is $M=(h/6)\mathrm{tridiag}(1,4,1)$ and the stiffness matrix is $K=h^{-1}\mathrm{tridiag}(-1,2,-1)$. The generalized eigenvalues are $6(1-\cos \theta )/[h^2(2+\cos \theta )]$. Applying forward Euler method requires $k\lambda \le 2$, giving the displayed uniform bound for $\mu =k/h^2$. For $J$ intervals with Dirichlet endpoints the exact finite-grid upper bound is $(2-\cos (\pi /J))/[3(1+\cos (\pi /J))]$, which approaches $1/6$. Replacing the consistent mass matrix by a lumped one changes the stability restriction.

For a nearly $D$-regular $s$-uniform hypergraph, sample each edge independently with probability $\alpha /D$, where $\alpha$ is small. Retain the isolated sampled edges as a matching in a hypergraph, and delete every vertex incident to any sampled edge before the next round. A vertex survives with probability $(1-\alpha /D{)}^{d(v)}\approx e^{-\alpha }$; a sampled edge meets another sampled edge with probability at most $s\alpha (1+o(1))$. Thus accepted edges cover order $\alpha$ of the vertices and collision waste is order ${\alpha }^2$. Small hypergraph codegrees control the overlap terms in residual-degree calculations. Repeating a bounded number of rounds while tracking near-regularity is the characteristic Rödl nibble mechanism.

For an isothermal gas slab, a horizontal Fourier mode has $\delta \rho =-ik\rho {\xi }_x-(\rho {\xi }_z{)}_z$, $W=c_s^2\delta \rho /\rho +\varphi$, ${\omega }^2{\xi }_x=ikW$, ${\omega }^2{\xi }_z=W_z$ and $({\partial }_z^2-k^2)\varphi =4\pi G\delta \rho$. Multiply momentum by $\rho {\mathbf{\xi }}^{\ast }$ and integrate by parts, then use the Poisson equation for Newtonian gravity. Vanishing pressure-work and gravitational boundary terms give the displayed real quadratic identity, with $I=\int \rho |\mathbf{\xi }{|}^{2}dz>0$. Thus every admissible squared eigenfrequency is real; a negative value gives gravitational growth rather than an oscillatory instability.

For a self-gravitating isothermal slab with $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ and $H^2={\sigma }^2/(2\pi G{\rho }_0)$, hydrostatic equilibrium gives the displayed Newtonian gravitational potential. A test particle has constant in-plane velocity and conserved vertical energy $E_z={\dot{z}}^2/2+\Phi (z)-\Phi (0)$. Its vertical motion is periodic for every finite $E_z>0$, with turning height $H\mathrm{arcosh}\exp [E_z/(2{\sigma }^2)]$. Near the plane it is a harmonic oscillator of squared frequency $4\pi G{\rho }_0$; at large height the force tends to the constant $2\pi G\Sigma$, where $\Sigma =2{\rho }_{0}H$. The ideal infinite slab has no finite vertical escape energy.

If $B$ is a positive riskless numéraire of finite variation, $S$ the ex-dividend price vector and $D$ its cash-dividend rate, a self-financing portfolio with risky holdings $\pi$ satisfies $d(X/B)=\pi \cdot dY$. This follows by differentiating $X/B$, substituting $dX=\varphi dB+\pi \cdot (dS+Ddt)$ and using $X=\varphi B+\pi \cdot S$. If $Y$ is a martingale under an equivalent measure, discounted wealth is a local martingale. A deterministic lower bound on discounted wealth makes it a supermartingale, ruling out arbitrage. The cumulative dividends here are discounted at their own payment times.

For $p(x,t)=xg(x,t)+B_{t}h(x,t)$ and deterministic $d{B}_t=\rho B_{t}dt$, Itô formula shows that a self-financing portfolio must have $p_x=g$. This is equivalent to $x{g}_x+B_t{h}_x=0$. Differentiating it gives $p_{xx}=g_x$, so the remaining gain identity is $x{g}_t+B_t{h}_t+{\sigma }^2{x}^2{g}_x/2=0$. Conversely these two identities give $dp=gdS+hdB$. A value PDE alone does not verify arbitrary prescribed holdings.

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.