Cauchy Condensation Test

Cauchy Condensation Test: Let (an)(a_n)(an) be a nonincreasing sequence of nonnegative real numbers. Then ∑n=1∞an converges ⟺∑k=0∞2ka2k converges.\sum_{n=1}^\infty a_n \text{ converges } \Longleftrightarrow \sum_{k=0}^\infty 2^k a_{2^k} \text{ converges}.∑n=1∞an converges ⟺∑k=0∞2ka2k converges. ...

Cauchy criterion for convergence in R^k

Cauchy criterion for convergence in Rk\mathbb{R}^kRk: A sequence (xn)(x_n)(xn) in Rk\mathbb{R}^kRk converges (with respect to the Euclidean norm ) if and only if it is a Cauchy sequence ; i.e., xn→x for some x∈Rk⟺∀ε>0 ∃N ∀m,n≥N: ∥xn−xm∥<ε.x_n\to x \text{ for some } x\in\mathbb{R}^k \quad\Longleftrightarrow\quad \forall \varepsilon>0\;\exists N\;\forall m,n\ge N:\ \|x_n-x_m\|<\varepsilon.xn→x for some x∈Rk⟺∀ε>0∃N∀m,n≥N: ∥xn−xm∥<ε. ...

Cauchy implies bounded

Cauchy implies bounded: Let (X,d)(X,d)(X,d) be a metric space and let (xn)(x_n)(xn) be a Cauchy sequence in XXX. Then (xn)(x_n)(xn) is bounded : there exist x∈Xx\in Xx∈X and R>0R>0R>0 such that d(xn,x)≤Rd(x_n,x)\le Rd(xn,x)≤R for all nnn. ...

Cauchy Mean Value Theorem

Cauchy Mean Value Theorem: Let f,g:[a,b]→Rf,g:[a,b]\to\mathbb{R}f,g:[a,b]→R be continuous on [a,b][a,b][a,b] and differentiable on (a,b)(a,b)(a,b). Then there exists c∈(a,b)c\in(a,b)c∈(a,b) such that (f(b)−f(a))g′(c)=(g(b)−g(a))f′(c). \bigl(f(b)-f(a)\bigr)g'(c)=\bigl(g(b)-g(a)\bigr)f'(c). (f(b)−f(a))g′(c)=(g(b)−g(a))f′(c). If moreover g(b)≠g(a)g(b)\neq g(a)g(b)=g(a) and g′(c)≠0g'(c)\neq 0g′(c)=0, then this can be rewritten as f′(c)g′(c)=f(b)−f(a)g(b)−g(a). \frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}. g′(c)f′(c)=g(b)−g(a)f(b)−f(a). ...

Cauchy product of two series

Let ∑n=0∞an\sum_{n=0}^\infty a_n∑n=0∞an and ∑n=0∞bn\sum_{n=0}^\infty b_n∑n=0∞bn be series in R\mathbb{R}R or C\mathbb{C}C. Their Cauchy product is the series ∑n=0∞cn\sum_{n=0}^\infty c_n∑n=0∞cn defined by ...

Cauchy sequence

Let (X,d)(X,d)(X,d) be a metric space and let (xn)(x_n)(xn) be a sequence in XXX. The sequence is Cauchy if ∀ε>0, ∃N∈N such that ∀m,n≥N, d(xm,xn)<ε.\forall \varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall m,n\ge N,\ d(x_m,x_n)<\varepsilon.∀ε>0, ∃N∈N such that ∀m,n≥N, d(xm,xn)<ε.A Cauchy sequence is one that has “intrinsic convergence” without reference to a candidate limit. Completeness of a metric space is the statement that every Cauchy sequence actually converges in the space. ...

Cauchy–Hadamard Theorem

Cauchy–Hadamard Theorem: Consider the power series ∑n=0∞an(x−x0)n \sum_{n=0}^\infty a_n (x-x_0)^n ∑n=0∞an(x−x0)n with an∈Ra_n\in\mathbb{R}an∈R or C\mathbb{C}C. Define L=lim sup⁡n→∞∣an∣n∈[0,∞]. L=\limsup_{n\to\infty}\sqrt[n]{|a_n|}\in[0,\infty]. L=limsupn→∞n∣an∣∈[0,∞]. Then the radius of convergence RRR is R=1L, R=\frac{1}{L}, R=L1, with the conventions 1/0=∞1/0=\infty1/0=∞ and 1/∞=01/\infty=01/∞=0. The series converges absolutely for ∣x−x0∣<R|x-x_0|<R∣x−x0∣<R and diverges for ∣x−x0∣>R|x-x_0|>R∣x−x0∣>R. ...

Cauchy–Schwarz inequality

Cauchy–Schwarz inequality: In an inner product space (V,⟨⋅,⋅⟩)(V,\langle\cdot,\cdot\rangle)(V,⟨⋅,⋅⟩), for all u,v∈Vu,v\in Vu,v∈V, ∣⟨u,v⟩∣≤∥u∥ ∥v∥,where ∥u∥=⟨u,u⟩. |\langle u,v\rangle|\le \|u\|\,\|v\|, \qquad \text{where } \|u\|=\sqrt{\langle u,u\rangle}. ∣⟨u,v⟩∣≤∥u∥∥v∥,where ∥u∥=⟨u,u⟩. Moreover, equality holds if and only if uuu and vvv are linearly dependent (i.e., one is a scalar multiple of the other). ...

Chain rule (multivariable)

Chain rule (multivariable): Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn and V⊆RmV\subseteq\mathbb{R}^mV⊆Rm be open. Suppose f:U→Vf:U\to Vf:U→V is differentiable at a∈Ua\in Ua∈U and g:V→Rpg:V\to\mathbb{R}^pg:V→Rp is differentiable at f(a)f(a)f(a). Then g∘f:U→Rpg\circ f:U\to\mathbb{R}^pg∘f:U→Rp is differentiable at aaa and D(g∘f)(a)=Dg(f(a))∘Df(a). D(g\circ f)(a)=Dg(f(a))\circ Df(a). D(g∘f)(a)=Dg(f(a))∘Df(a). In matrix form (with Jacobians ), Jg∘f(a)=Jg(f(a)) Jf(a). J_{g\circ f}(a)=J_g(f(a))\,J_f(a). Jg∘f(a)=Jg(f(a))Jf(a). ...

Change of variables (coordinate transformation) for multiple integrals

A change of variables for a multiple integral refers to using a coordinate map Φ\PhiΦ to rewrite an integral over a region in Rn\mathbb{R}^nRn. Typically, one considers: open sets U,V⊆RnU,V\subseteq \mathbb{R}^nU,V⊆Rn, a C1C^1C1 bijection Φ:U→V\Phi:U\to VΦ:U→V with C1C^1C1 inverse (often a diffeomorphism ), and a region E⊆UE\subseteq UE⊆U whose image is Φ(E)⊆V\Phi(E)\subseteq VΦ(E)⊆V. The associated Jacobian determinant is det⁡DΦ(u)\det D\Phi(u)detDΦ(u), and the change-of-variables formula (a theorem stated separately) relates ∫Φ(E)f(x) dx\int_{\Phi(E)} f(x)\,dx∫Φ(E)f(x)dx to an integral over EEE involving f(Φ(u))∣det⁡DΦ(u)∣f(\Phi(u))|\det D\Phi(u)|f(Φ(u))∣detDΦ(u)∣. ...