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The Liouville approximation theorem states that an irrational algebraic number $\alpha$ of degree $d\ge 2$ satisfies $|\alpha -a/b|\ge C_{\alpha }{b}^{-d}$ for all integers $a$ and $b\ge 1$, for some $C_{\alpha }>0$. The constant is independent of the rational approximation.

The required condition is infinitely many of the coefficients equal one. If only finitely many do, the series is a finite sum of rational numbers. Conversely let $\alpha$ be the sum with infinitely many nonzero terms. Its truncation $s_N$ has denominator dividing $q_N=10^{N!}$. For $N\ge 2$, its strictly positive tail satisfiesIf $\alpha =a/b$ were rational, the distinct fractions $\alpha ,s_N$ would have distance at least $1/(b{q}_N)$, contradicting the tail bound for large $N$. Thus $\alpha$ is irrational. If it were algebraic of fixed degree $d$, apply the Liouville approximation theorem to $s_N$. Even if its reduced denominator is smaller than $q_N$, the lower bound is at least $C_{\alpha }{q}_N^{-d}$. The displayed upper bound contradicts this as $N\to \infty$. HenceThe coincidence $0!=1!$ changes only the first rational contribution and has no effect on this criterion.

The microcanonical ensemble assigns equal probability to accessible quantum states in a narrow energy shell of an isolated system with fixed energy, particle number and external parameters. To derive the canonical ensemble, put a small system in weak thermal contact with a much larger isolated bath. If the small-system state has energy $E_i$, its probability is proportional to the number ${\Omega }_b(E_{\mathrm{total}}-E_i)$ of bath states. Expanding the bath entropy gives $S_b(E_{\mathrm{total}}-E_i)=S_b(E_{\mathrm{total}})-E_i/T+\cdots$, since $\partial S_b/\partial E=1/T$. Negligible interaction energy and an effectively constant bath temperature therefore give $p_i=e^{-E_i/(k_{B}T)}/Z$. The microcanonical ensemble suits an isolated fixed-energy system; the canonical ensemble suits a system exchanging energy with a thermostat at fixed temperature, with particle number still fixed.

For one atom, summing the finite geometric series over $m=-J,\dots ,J$ gives the partition functionAt $x=0$ the ratio is understood by continuity as $2J+1$. Logarithmic differentiation gives the mean magnetic dipole momentThis is odd in $x$, has small-$x$ expansion ${\mu }_{B}J(J+1)x/3+O(x^3)$, and saturates at $\pm J{\mu }_B$. The requested common graph is plotted in units of ${\mu }_B$: its ordinate is $⟨{\mu }_z⟩/(J{\mu }_B)$, so multiplication by ${\mu }_B$ gives $⟨{\mu }_z⟩/J$. The normalized slopes at zero are $(J+1)/3$ and the limits are $\pm 1$.

The magnetic susceptibility isEquivalently it is $N{\mu }_B^2\mathrm{Var}(m)/(k_{B}T)\ge 0$. For $k_{B}T\gg {\mu }_B|B|$, the apparent singular terms cancel, yielding Curie lawFor fixed nonzero $B$, $T\to 0$ and integer $J\ge 1$, the first excited magnetic level is separated from the aligned ground state by ${\mu }_B|B|$. Its exponentially small occupation givesThe moments are saturated, so changing the magnetic field hardly changes their alignment. This is a fixed-nonzero-field low-temperature limit. At exactly $B=0$, differentiating first gives the Curie expression for every $T>0$, which diverges as $T\to 0$; the two limits do not commute in this noninteracting model. For $J=0$ both moment and susceptibility vanish identically.

The exponent $\Phi (t)=\frac{i}{2}(t-t^{-1})$ has saddles $t=\pm i$, with $\Phi (-i)=1$ and $\Phi (i)=-1$. Since the integrand is analytic off the origin, deform the contour to the unit circle and rotate $t=-i{e}^{i\theta }$. This gives the exact representationThe dominant saddle at $\theta =0$ is a steepest-descent direction for the method of steepest descent: $\cos \theta =1-{\theta }^2/2+{\theta }^4/24-\cdots$. Set $\theta =s/\sqrt{x}$ and expand the analytic amplitude against $e^{-s^2/2}$. Integrating its Gaussian moments gives the leading factor $e^x/\sqrt{2\pi x}$; the rest of the contour is exponentially smaller than this dominant contribution outside any fixed neighbourhood of the maximum.

To obtain every coefficient efficiently, the integral satisfies $x^2{I}_0^{\prime \prime }+x{I}_0^{\prime }-x^2{I}_0=0$, by differentiating and integrating the derivative of $\sin \theta e^{x\cos \theta }$. Substitute $I_0=e^x{x}^{-1/2}\sum a_n{x}^{-n}/\sqrt{2\pi }$ to get $a_0=1$ and $a_n=(2n-1{)}^2{a}_{n-1}/(8n)$. Therefore the full asymptotic expansion isHere $(-1)!!=1$. For each fixed truncation at $n=N$, the relative remainder is $O(x^{-N-1})$ as $x\to +\infty$. The infinite series is generally divergent, and this all-orders expansion does not purport to resolve exponentially small saddle contributions or their Stokes phenomenon.

With the sign convention of the displayed operator, the Lax equation is $L_t=[L,A]$. Differentiating $L\phi =k^2\phi$ shows that ${\phi }_t+A\phi$ is another solution at the same spectral parameter. The normalization at $-\infty$ fixessince $A{e}^{-ikx}=4i{k}^3{e}^{-ikx}$ there. At $+\infty$, comparison of the two plane-wave coefficients givesThus the transmission denominator $a$ is time independent. This is the Time evolution of KdV scattering data for the stated normalization.

The logarithmic derivative $S={\phi }_x/\phi +ik$ obeys the Riccati equationIts formal large-$k$ expansion givesThere is an important qualification to the integral identity in the question. For real $k$ with nonzero reflection, $e^{ikx}\phi \sim a+b{e}^{2ikx}$ has no limit at $+\infty$, so the literal improper integral of the exact $S$ need not exist. Continue to $\mathrm{Im}k>0$, where the reflected exponential decays relative to $e^{-ikx}$, and take sufficiently large $k$ away from zeros and with a consistent logarithm. Then the normalization givesThe second equality is an asymptotic expansion, not an asserted convergent series. Time independence of $a$ fixes every coefficient, so each $\int S_{n}dx$ is a conserved quantity of the Korteweg-De Vries equation. The first two nontrivial examples are $-\int udx$ and $\int u^{2}dx$.

For the usual real-valued KdV potential and real formal $k$, write $S=X+iY$ with real coefficient functions. Its imaginary part of the Riccati equation is $Y^{\prime }-2kX+2XY=0$, henceIn the formal expansion, $X$ contains precisely the even-indexed $S_n$ and $Y$ the odd-indexed ones. Rapid decay of $u$ and its derivatives makes every coefficient of the logarithm vanish at both spatial ends. Therefore $\boxed{{\int }_{\mathbb{R}}{S}_{2j}dx=0(j\ge 1)}$. This is a coefficientwise formal identity; it does not claim that the exact real-axis reflected solution has a convergent integral of $X$. The analytically continued scattering identity and the formal expansion are the conventions needed to make the requested argument valid.

A Cauchy sequence is a sequence $(x_n)$ such that for every $\epsilon >0$ there is $N$ for whichThe general principle of convergence, or completeness of the real numbers, states that a real sequence converges if and only if it is Cauchy.

For the final claim, let $m=\lfloor n/2\rfloor$. Since $(a_n)$ is decreasing and positive,Because the series $\sum a_k$ converges, its tails tend to zero. Also $n-m\ge n/2$, soThe squeeze theorem gives

The comparison test says that if $0\le b_n\le c_n$ eventually and $\sum c_n$ converges, then $\sum b_n$ converges. If $\sum a_n{z}_0^n$ converges, its terms are bounded: $|a_n{z}_0^n|\le M$. For $|z_1|<|z_0|$,so comparison with a geometric series proves absolute convergence.

The radius $R$ is the number for which the power series converges absolutely for $|z|<R$ and diverges for $|z|>R$. Given $r_i<R_i$, both $\sum |a_n|r_1^n$ and $\sum |b_n|r_2^n$ converge, so their terms are bounded, say $|a_n|r_1^n\le M$. ThenThus $R\ge r_1{r}_2$; letting $r_i\uparrow R_i$ gives $R\ge R_1{R}_2$.

Another application of the Cauchy-Schwarz inequality givesWhen $p>1/2$, the second factor is bounded because $2p>1$, so the series converges.

At the endpoint, take $a_1=1$ and, for $n\ge 2$,Then $\sum a_n$ converges, whilewhose series diverges. This proves both parts of weighted square roots of a summable sequence.

Suppose the series converges at $w\ne 0$. Then its terms $a_n{w}^n$ are bounded, say by $M$. Whenever $|z|<|w|$,so comparison with a geometric series proves absolute convergence at $z$.

Letwith the natural values $0$ or $\infty$ allowed. If $|z|<R$, choose a convergent point $w$ with $|z|<|w|\le R$; the preceding argument gives convergence at $z$. If $|z|>R$, convergence at $z$ would contradict the definition of $R$. Hence the series converges for $|z|<R$ and diverges for $|z|>R$. No universal assertion is possible on $|z|=R$.

The ratio test says that ifthen $\sum a_n$ converges absolutely. If the ratio has a limit $L>1$, the series diverges.

For the first assertion, choose $q$ with $L<q<1$. For all sufficiently large $n$,Iteration bounds the tail by a constant multiple of the convergent geometric series $\sum q^n$. If $L>1$, choose $q$ with $1<q<L$. Eventually $|a_{n+1}|\ge q|a_n|$, so $a_n$ cannot tend to zero; the series therefore diverges.

The homogeneous characteristic equation isA linear particular solution $u_n=an+b$ givesso $a=-1/4$ and $b=-11/16$. ThusThe initial data yield $A=1$ and $B=-5/16$, hence

For the ordinary generating function of a recurrence, multiply by $x^n$ and sum for $n\ge 2$. Using $u_0=0,u_1=2$ givesEquivalently,Expanding each geometric series gives exactly the displayed formula for $u_n$, verifying consistency.

Conditioning on $N$ and using the moment-generating function of a unit exponential variable,where the geometric series converges precisely when $\theta <p$. This is the moment-generating function of an exponential variable of rate $p$, so the geometric sum of exponential variables satisfies

A covering map is locally a disjoint union of homeomorphisms onto the base. A space is simply connected when it is path-connected and every loop is null-homotopic. By the uniformization theorem, the simply connected Riemann surfaces areup to analytic isomorphism.

A lattice is a discrete subgroupwith ${\omega }_1/{\omega }_2\notin \mathbb{R}$. The Weierstrass elliptic functionconverges normally on compact subsets of $\mathbb{C}\setminus L$, defines a meromorphic map $\mathbb{C}\to {\mathbb{C}}_{\infty }$, and is nonconstant because it has a double pole at every lattice point. Reindexing the normally convergent derivative series shows that ${\wp }_L^{\prime }$ is $L$-periodic; evenness fixes the integration constants, so ${\wp }_L(z+\omega )={\wp }_L(z)$.

The invariance makeswell-defined and unique on $\mathbb{C}/L$. It is analytic because the quotient projection is locally biholomorphic.

Neither map is a covering. For any nonzero $\omega \in L$, periodicity and oddness of ${\wp }_L^{\prime }$ giveso ${\wp }_L^{\prime }(\omega /2)=0$. The map is not a local homeomorphism there; since the quotient projection is locally biholomorphic, the descended map fails for the same reason.

There is no covering $\mathbb{C}/L\to {\mathbb{C}}_{\infty }$. The torus is connected, while every connected covering of the simply connected sphere is a homeomorphism. A torus is not homeomorphic to a sphere, for example because their fundamental groups are ${\mathbb{Z}}^2$ and $0$.

The Markov property at deterministic times givesThus $(Z_n)$ is a discrete-time Markov chain with transition matrixwhen the minimal chain is nonexplosive, $P(h)=e^{hQ}$.

For an irreducible chain, state $i$ is recurrent in continuous time exactly whenand it is recurrent for the skeleton exactly when ${\sum }_{n\ge 0}{p}_{ii}(nh)=\infty$. If $t\in [nh,(n+1)h]$, the event of staying at $i$ suppliesIntegrating over each interval shows that the integral diverges exactly when the series does. Irreducibility then makes recurrence equivalent for the two chains. This is recurrence equivalence for a CTMC and its fixed-time skeleton.

The matrix exponential is the absolutely convergent seriesMultiplying the two expansions givesOn the other hand,Subtracting yieldsThis is the leading Lie-Trotter splitting commutator error.