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Question Number 214111 Answers: 0 Comments: 0

hmmmm............ can we find closed form of sum Σ_(j=0) ^∞ (1/2^2^j ) ?? or any idea..?

$$\mathrm{hmmmm}............ \\ $$$$\mathrm{can}\:\mathrm{we}\:\mathrm{find}\:\mathrm{closed}\:\mathrm{form}\:\mathrm{of}\:\mathrm{sum} \\ $$$$\underset{{j}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}^{{j}} } }\:??\:\mathrm{or}\:\mathrm{any}\:\mathrm{idea}..? \\ $$

Question Number 214104 Answers: 0 Comments: 0

Question Number 214100 Answers: 2 Comments: 0

Question Number 214093 Answers: 2 Comments: 1

Question Number 214091 Answers: 1 Comments: 0

Question Number 214072 Answers: 4 Comments: 0

Question Number 214140 Answers: 0 Comments: 0

Let F ; R^n →R^n be continuously differentiable a) assume that the Jacoboian matrix (∂f_i /∂x_j ) has rank n everywhere prove that f(R^n )is open b) suppose that f^(−1) (K) is compact whenever K⊂R^n is compact. prove that f(R^n ) is closed

$$\mathrm{Let}\:{F}\:;\:\mathbb{R}^{{n}} \rightarrow\mathbb{R}^{{n}} \:\mathrm{be}\:\mathrm{continuously} \\ $$$$\mathrm{differentiable}\: \\ $$$$\left.\boldsymbol{\mathrm{a}}\right)\:\mathrm{assume}\:\mathrm{that}\:\mathrm{the}\:\mathrm{Jacoboian}\:\mathrm{matrix}\: \\ $$$$\frac{\partial{f}_{{i}} }{\partial{x}_{{j}} }\:\mathrm{has}\:\mathrm{rank}\:{n}\:\mathrm{everywhere} \\ $$$$\mathrm{prove}\:\mathrm{that}\:{f}\left(\mathbb{R}^{{n}} \right)\mathrm{is}\:\mathrm{open} \\ $$$$\left.\boldsymbol{\mathrm{b}}\right)\:\mathrm{suppose}\:\mathrm{that}\:{f}^{−\mathrm{1}} \left({K}\right)\:\mathrm{is}\:\mathrm{compact} \\ $$$$\mathrm{whenever}\:{K}\subset\mathbb{R}^{{n}} \:\mathrm{is}\:\mathrm{compact}. \\ $$$$\mathrm{prove}\:\mathrm{that}\:{f}\left(\mathbb{R}^{{n}} \right)\:\mathrm{is}\:\mathrm{closed} \\ $$

Question Number 214068 Answers: 0 Comments: 0

Question Number 214065 Answers: 0 Comments: 1

Exercice pratique: determiner Volume de la forme AMBN :est dans un plan horisontale ci−dessous.(OI=12) (ACB) assimile a une chainette

$$\boldsymbol{{Exercice}}\:\boldsymbol{{pratique}}: \\ $$$$\mathrm{determiner}\:\mathrm{Volume}\:\mathrm{de}\:\mathrm{la}\:\mathrm{forme} \\ $$$$\mathrm{AMBN}\::\mathrm{est}\:\mathrm{dans}\:\mathrm{un}\:\mathrm{plan}\:\mathrm{horisontale} \\ $$$$\mathrm{ci}−\mathrm{dessous}.\left(\mathrm{OI}=\mathrm{12}\right) \\ $$$$\left(\mathrm{ACB}\right)\:\mathrm{assimile}\:\mathrm{a}\:\mathrm{une}\:\mathrm{chainette} \\ $$

Question Number 214059 Answers: 2 Comments: 0

Question Number 214098 Answers: 1 Comments: 0

Let F be Field of characteristic 0 L_i (i=1,2) be two algebraic extension of F , and L_1 L_2 be a field in F^ (where F^ is the algebraic closure of F) defined by {l_1 l_2 ∣l_i ∈L_i (i=1,2)} 1. show that if L_1 and L_2 are galois over F then L_1 L_2 is also Galois over F 2. show that if G(L_1 /F^ ) and G(L_2 /F^ ) are Solvable , then Gal(L_1 L_2 /F^ ) is also Solvable

$$\mathrm{Let}\:{F}\:\mathrm{be}\:\:\mathrm{Field}\:\mathrm{of}\:\mathrm{characteristic}\:\mathrm{0} \\ $$$${L}_{{i}} \:\left({i}=\mathrm{1},\mathrm{2}\right)\:\mathrm{be}\:\mathrm{two}\:\mathrm{algebraic}\:\mathrm{extension} \\ $$$$\mathrm{of}\:{F}\:,\:\mathrm{and}\:{L}_{\mathrm{1}} {L}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{field}\:\mathrm{in}\:\bar {{F}}\: \\ $$$$\left(\mathrm{where}\:\bar {{F}}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{algebraic}\:\mathrm{closure}\:\:\mathrm{of}\:{F}\right) \\ $$$$\mathrm{defined}\:\mathrm{by}\:\left\{{l}_{\mathrm{1}} {l}_{\mathrm{2}} \mid{l}_{{i}} \in{L}_{{i}} \:\left({i}=\mathrm{1},\mathrm{2}\right)\right\} \\ $$$$\mathrm{1}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:{L}_{\mathrm{1}} \:\mathrm{and}\:{L}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{galois}\:\mathrm{over}\:{F} \\ $$$$\mathrm{then}\:{L}_{\mathrm{1}} {L}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{also}\:\mathrm{Galois}\:\mathrm{over}\:{F} \\ $$$$\mathrm{2}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:{G}\left({L}_{\mathrm{1}} /{F}^{\:} \right)\:\mathrm{and}\:{G}\left({L}_{\mathrm{2}} /{F}^{\:} \right) \\ $$$$\mathrm{are}\:\mathrm{Solvable}\:,\:\mathrm{then}\:\mathrm{Gal}\left({L}_{\mathrm{1}} {L}_{\mathrm{2}} /{F}^{\:} \right)\:\mathrm{is}\:\mathrm{also} \\ $$$$\mathrm{Solvable} \\ $$

Question Number 214097 Answers: 1 Comments: 0

Question Number 214051 Answers: 1 Comments: 1

evaluate ∫_0 ^( π) e^(sin^2 (u)) du... i use Feynman′s trick to solve integral ∫_0 ^( π) e^(sin^2 (u)) du=I I(t)=∫_0 ^( π) e^(tsin^2 (u)) du I^((1)) (t)=∫_0 ^( π) sin^2 (u)e^(tsin^2 (u)) du ∫_0 ^( π) (1−cos^2 (u))e^(t∙sin^2 (u)) du ∫_0 ^( π) e^(t∙sin^2 (u)) du−∫_0 ^( π) cos^2 (u)e^(t∙sin^2 (u)) du ∫_0 ^( π) e^(t∙sin^2 (u)) du−∫_0 ^( π) sin^2 (u+(π/2))e^(t∙sin^2 (u)) du u+(π/2) =w → du=dw ∫_(π/2) ^((3π)/2) sin^2 (w)e^(t∙cos^2 (w)) dw= ∫_(π/2) ^( ((3π)/2)) sin^2 (w)e^(t(1−sin^2 (w))) dw ∫_(π/2) ^((3π)/2) sin^2 (w)e^t e^(−tsin^2 (w)) dw=e^t ∫_(π/2) ^((3π)/2) sin^2 (w)e^(−tsin^2 (w)) dw I^((1)) (t)=∫_0 ^( π) e^(t∙sin^2 (u)) du−e^t ∫_0 ^( π) sin^2 (u)e^(−t∙sin^2 (u)) du I can′t ...anymore...help

Question Number 214050 Answers: 0 Comments: 0

If A: a_1 , a_2 , ..., a_(62) and B: b_1 , b_2 , ..., b_(62) are two strictly increasing natural number sequences such that a_(62) ≤755 and b_(62) ≤755. Find the maximum of Σ_(i=1) ^(62) ∣a_i −b_i ∣−∣Σ_(i=1) ^(62) (a_i −b_i )∣.

$$\mathrm{If}\:{A}:\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:...,\:{a}_{\mathrm{62}} \:\mathrm{and}\:{B}:\:{b}_{\mathrm{1}} ,\:{b}_{\mathrm{2}} ,\:...,\:{b}_{\mathrm{62}} \:\mathrm{are}\:\mathrm{two} \\ $$$$\mathrm{strictly}\:\mathrm{increasing}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{sequences} \\ $$$$\mathrm{such}\:\mathrm{that}\:{a}_{\mathrm{62}} \leqslant\mathrm{755}\:\mathrm{and}\:{b}_{\mathrm{62}} \leqslant\mathrm{755}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{of}\:\underset{{i}=\mathrm{1}} {\overset{\mathrm{62}} {\sum}}\mid{a}_{{i}} −{b}_{{i}} \mid−\mid\underset{{i}=\mathrm{1}} {\overset{\mathrm{62}} {\sum}}\left({a}_{{i}} −{b}_{{i}} \right)\mid. \\ $$

Question Number 214047 Answers: 1 Comments: 1