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Question Number 10664 Answers: 0 Comments: 0

S=Σ_(n∉P_(n≥1) ) ^∞ n Q=Σ_(n∈P_(n≥1) ) ^∞ n Prove if true: S>Q

$${S}=\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\notin\mathbb{P}}} {\overset{\infty} {\sum}}{n} \\ $$$${Q}=\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\in\mathbb{P}}} {\overset{\infty} {\sum}}{n} \\ $$$$\: \\ $$$$\mathrm{Prove}\:\mathrm{if}\:\mathrm{true}: \\ $$$${S}>{Q} \\ $$

Question Number 10662 Answers: 0 Comments: 0

determine if: (1/2^s )+(1/3^s )+(1/5^s )+...≥(1/1^s )+(1/4^s )+(1/6^s )+... or: Σ_(n∈P_(n≥1) ) ^∞ (1/n^s )≥Σ_(n∉P_(n≥1) ) ^∞ (1/n^s ), Re(s)>1 note: Σ_(n∈P_(n≥1) ) ^∞ (1/n^s )+Σ_(n∉P_(n≥1) ) ^∞ (1/n^s )=ζ(s)

$$\mathrm{determine}\:\mathrm{if}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}^{{s}} }+\frac{\mathrm{1}}{\mathrm{3}^{{s}} }+\frac{\mathrm{1}}{\mathrm{5}^{{s}} }+...\geqslant\frac{\mathrm{1}}{\mathrm{1}^{{s}} }+\frac{\mathrm{1}}{\mathrm{4}^{{s}} }+\frac{\mathrm{1}}{\mathrm{6}^{{s}} }+... \\ $$$$\mathrm{or}: \\ $$$$\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\in\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} }\geqslant\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\notin\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} },\:\:\:\:\:\:\:\:\:\:\mathrm{Re}\left({s}\right)>\mathrm{1} \\ $$$$\: \\ $$$$\mathrm{note}: \\ $$$$\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\in\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} }+\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\notin\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} }=\zeta\left({s}\right) \\ $$

Question Number 10660 Answers: 2 Comments: 0

ind the centre and radius of these: (a) 6x^2 +6y^2 −4x−5y−2=0 (b) x^2 +y^2 +6x+8y−1=0 (c) 3x^2 +3y^2 −4x+8y−2=0

$${ind}\:{the}\:{centre}\:{and}\:{radius}\:{of}\:{these}: \\ $$$$\left({a}\right)\:\mathrm{6}{x}^{\mathrm{2}} +\mathrm{6}{y}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{5}{y}−\mathrm{2}=\mathrm{0} \\ $$$$\left({b}\right)\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{8}{y}−\mathrm{1}=\mathrm{0} \\ $$$$\left({c}\right)\:\mathrm{3}{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{8}{y}−\mathrm{2}=\mathrm{0} \\ $$

Question Number 10656 Answers: 1 Comments: 0

find the equation of the circle with diameter AB where A is at (2,4) and B is at (−1,6)

$${find}\:{the}\:{equation}\:{of}\:{the}\:{circle}\:{with}\:{diameter}\:{AB}\:{where}\:{A}\:{is}\:{at}\:\left(\mathrm{2},\mathrm{4}\right)\:{and}\:{B}\:{is}\:{at}\:\left(−\mathrm{1},\mathrm{6}\right) \\ $$

Question Number 10655 Answers: 0 Comments: 0

Show me your favourite proofs relating to e

$$\mathrm{Show}\:\mathrm{me}\:\mathrm{your}\:\mathrm{favourite}\:\mathrm{proofs} \\ $$$$\mathrm{relating}\:\mathrm{to}\:{e} \\ $$

Question Number 10654 Answers: 0 Comments: 0

Show me your favorite proofs relating to π

$$\mathrm{Show}\:\mathrm{me}\:\mathrm{your}\:\mathrm{favorite}\:\mathrm{proofs} \\ $$$$\mathrm{relating}\:\mathrm{to}\:\pi \\ $$

Question Number 10653 Answers: 1 Comments: 0

A ship leaves a port P which lies in latitude 20°N. It sails due east through 30° of longitude and then through south to Q which lies on the equator. Calculate the distance it has travelled, (Take the cicumference of the earth to be 40,000 km). on the return jouney it sails due west through 30° of longitude and then due north back to P. Show that the difference in length between the outward and return jouney is approximately 201 kilometers. Using this value of 201 km and taking 1 knot to be 1.852 km/hr . Calculate the difference in time between the two journey assuming that on each journey the ship sails at an average speed of 25 knots. Really need help here sirs. God will always increase your wisdom.

Question Number 10640 Answers: 2 Comments: 0

A circular disc of mass 20kg and radius 15cm is mounted in an horizontal cylinder axel of radius 1.5cm, Calculate the kinectic energy of the disc after 1.2 secs if a force of 12N is applied tangentially to the axel.

$$\mathrm{A}\:\mathrm{circular}\:\mathrm{disc}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{20kg}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{15cm}\:\mathrm{is}\:\mathrm{mounted}\:\mathrm{in}\:\mathrm{an} \\ $$$$\mathrm{horizontal}\:\mathrm{cylinder}\:\mathrm{axel}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{1}.\mathrm{5cm},\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{kinectic} \\ $$$$\mathrm{energy}\:\mathrm{of}\:\mathrm{the}\:\mathrm{disc}\:\mathrm{after}\:\mathrm{1}.\mathrm{2}\:\mathrm{secs}\:\mathrm{if}\:\mathrm{a}\:\mathrm{force}\:\mathrm{of}\:\mathrm{12N}\:\mathrm{is}\:\mathrm{applied}\:\mathrm{tangentially} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{axel}. \\ $$

Question Number 10639 Answers: 1 Comments: 3

A manometer wire of lenght 60 cm is maintained under a tension of value 20V and an a.c is passed through the wire. If the density of the wire is 4000kgm^(−3) and it diamerter is 2mm. Calculate the frequency of the a.c

$$\mathrm{A}\:\mathrm{manometer}\:\mathrm{wire}\:\mathrm{of}\:\mathrm{lenght}\:\mathrm{60}\:\mathrm{cm}\:\mathrm{is}\:\mathrm{maintained}\:\mathrm{under} \\ $$$$\mathrm{a}\:\mathrm{tension}\:\mathrm{of}\:\mathrm{value}\:\mathrm{20V}\:\mathrm{and}\:\mathrm{an}\:\mathrm{a}.\mathrm{c}\:\mathrm{is}\:\mathrm{passed}\:\mathrm{through} \\ $$$$\mathrm{the}\:\mathrm{wire}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{density}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wire}\:\mathrm{is}\:\mathrm{4000kgm}^{−\mathrm{3}} \:\mathrm{and}\:\mathrm{it}\: \\ $$$$\mathrm{diamerter}\:\mathrm{is}\:\mathrm{2mm}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{frequency}\:\mathrm{of}\:\mathrm{the}\:\mathrm{a}.\mathrm{c} \\ $$

Question Number 10638 Answers: 0 Comments: 0

S=(√(1+(√(a+(√(a^2 +(√(a^3 +(√(...)))))))))), a>0

$${S}=\sqrt{\mathrm{1}+\sqrt{{a}+\sqrt{{a}^{\mathrm{2}} +\sqrt{{a}^{\mathrm{3}} +\sqrt{...}}}}},\:\:\:\:{a}>\mathrm{0} \\ $$

Question Number 10627 Answers: 1 Comments: 0

60^(15) ×30^8 ×35^6 ×60^(15) =?

$$\mathrm{60}^{\mathrm{15}} ×\mathrm{30}^{\mathrm{8}} ×\mathrm{35}^{\mathrm{6}} ×\mathrm{60}^{\mathrm{15}} =? \\ $$

Question Number 10621 Answers: 1 Comments: 0

Question Number 10620 Answers: 1 Comments: 1

Question Number 10612 Answers: 2 Comments: 1

Question Number 10607 Answers: 1 Comments: 0

Solve for x: 2x^3 + 2x^2 − 5x − 1 = 0

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}: \\ $$$$\mathrm{2x}^{\mathrm{3}} \:+\:\mathrm{2x}^{\mathrm{2}} \:−\:\mathrm{5x}\:−\:\mathrm{1}\:=\:\mathrm{0} \\ $$

Question Number 10597 Answers: 1 Comments: 0

prove that ∣z_1 − z_2 ∣ ≤ ∣z_1 ∣ + ∣z_2 ∣

$$\mathrm{prove}\:\mathrm{that} \\ $$$$\mid\mathrm{z}_{\mathrm{1}} −\:\mathrm{z}_{\mathrm{2}} \mid\:\leqslant\:\mid\mathrm{z}_{\mathrm{1}} \mid\:+\:\mid\mathrm{z}_{\mathrm{2}} \mid \\ $$

Question Number 10583 Answers: 2 Comments: 3

The sum of the 4^(th ) and 6^(th ) terms of an AP is 42. the sum of the third and 9th terms of the proression is 52. Find the first term , the common difference and the sum of the first 10 terms of the progression.

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{4}^{\mathrm{th}\:} \:\mathrm{and}\:\mathrm{6}^{\mathrm{th}\:} \mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{is}\:\mathrm{42}.\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{third}\:\mathrm{and}\:\mathrm{9th}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{proression}\:\mathrm{is}\:\mathrm{52}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{term}\:,\:\mathrm{the}\:\mathrm{common}\:\mathrm{difference}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first} \\ $$$$\mathrm{10}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{progression}. \\ $$

Question Number 10577 Answers: 1 Comments: 0

5^(71) +5^(72) +5^(73) =?

$$\mathrm{5}^{\mathrm{71}} +\mathrm{5}^{\mathrm{72}} +\mathrm{5}^{\mathrm{73}} =? \\ $$

Question Number 10575 Answers: 1 Comments: 1

3(2^2 +1)(2^4 +1)(2^8 +1)(2^(16) +1)(2^(32) +1)(2^(64) +1)(2^(128) +1)=...?

$$\mathrm{3}\left(\mathrm{2}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{4}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{8}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{16}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{32}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{64}} +\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{128}} +\mathrm{1}\right)=...? \\ $$

Question Number 10572 Answers: 1 Comments: 0

factorise x^3 + 1

$$\mathrm{factorise} \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1} \\ $$

Question Number 10571 Answers: 0 Comments: 0

Question Number 10570 Answers: 0 Comments: 0

n(A)=12 n(B)=10 ⇒min(n(A−B−B^′ ))

$${n}\left({A}\right)=\mathrm{12} \\ $$$${n}\left({B}\right)=\mathrm{10} \\ $$$$\Rightarrow{min}\left({n}\left({A}−{B}−{B}^{'} \right)\right) \\ $$

Question Number 10566 Answers: 2 Comments: 0

Question Number 10564 Answers: 1 Comments: 0

Prove that: lim_(ε→0) ((−1+x^ε )/ε) = ln(x)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\epsilon\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{1}+{x}^{\epsilon} }{\epsilon}\:=\:\mathrm{ln}\left({x}\right) \\ $$

Question Number 10563 Answers: 0 Comments: 0

can someone explain to me big K notation? (I don′t know the name) It is related to continuous fractions. e.g. x=b_0 +K_(i=1) ^∞ (a_i /b_i ) e^x =(x^0 /(0!))+(x^1 /(1!))+(x^2 /(2!))+... e^x =Σ_(i=0) ^∞ (x^i /(i!)) =1+(x/(1−((1x)/(2+x−((2x)/(3+x−((3x)/(...)))))))) How do you interperate this in big K notation?

$$\mathrm{can}\:\mathrm{someone}\:\mathrm{explain}\:\mathrm{to}\:\mathrm{me} \\ $$$$\mathrm{big}\:\mathrm{K}\:\mathrm{notation}?\:\left(\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{the}\:\mathrm{name}\right) \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{related}\:\mathrm{to}\:\mathrm{continuous}\:\mathrm{fractions}. \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\:{x}={b}_{\mathrm{0}} +\underset{{i}=\mathrm{1}} {\overset{\infty} {\boldsymbol{\mathrm{K}}}}\frac{{a}_{{i}} }{{b}_{{i}} } \\ $$$$\: \\ $$$${e}^{{x}} =\frac{{x}^{\mathrm{0}} }{\mathrm{0}!}+\frac{{x}^{\mathrm{1}} }{\mathrm{1}!}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}!}+... \\ $$$${e}^{{x}} =\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{i}} }{{i}!} \\ $$$$\:\:\:\:\:=\mathrm{1}+\frac{{x}}{\mathrm{1}−\frac{\mathrm{1}{x}}{\mathrm{2}+{x}−\frac{\mathrm{2}{x}}{\mathrm{3}+{x}−\frac{\mathrm{3}{x}}{...}}}} \\ $$$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{interperate}\:\mathrm{this}\:\mathrm{in} \\ $$$$\mathrm{big}\:\mathrm{K}\:\mathrm{notation}? \\ $$

Question Number 10562 Answers: 0 Comments: 0

x= [(x_1 ),(x_2 ),(( ⋮)),(x_n ) ] y= [(y_1 ),(y_2 ),(( ⋮)),(y_n ) ] x, y ∈ R^n 1. Prove the length of the vector x, denoted ∣∣x∣∣, is equal to (√(x_1 ^2 +x_2 ^2 +...+x_n ^2 )) 2. Determine if ∣∣x−y∣∣=∣∣y−x∣∣

$$\boldsymbol{{x}}=\begin{bmatrix}{{x}_{\mathrm{1}} }\\{{x}_{\mathrm{2}} }\\{\:\vdots}\\{{x}_{{n}} }\end{bmatrix}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{y}}=\begin{bmatrix}{{y}_{\mathrm{1}} }\\{{y}_{\mathrm{2}} }\\{\:\vdots}\\{{y}_{{n}} }\end{bmatrix}\:\:\:\:\:\:\:\:\boldsymbol{{x}},\:\boldsymbol{{y}}\:\in\:\mathbb{R}^{{n}} \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{vector}\:\boldsymbol{{x}},\:\mathrm{denoted}\:\mid\mid\boldsymbol{{x}}\mid\mid, \\ $$$$\:\:\:\:\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\sqrt{{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +...+{x}_{{n}} ^{\mathrm{2}} } \\ $$$$\mathrm{2}.\:\mathrm{Determine}\:\mathrm{if}\:\:\:\mid\mid\boldsymbol{{x}}−\boldsymbol{{y}}\mid\mid=\mid\mid\boldsymbol{{y}}−\boldsymbol{{x}}\mid\mid \\ $$

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