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Question Number 149342 Answers: 1 Comments: 1

is this true and is there a proof? ∀k∈N∣k>1 ((Σ_(n=1) ^(k^2 −1) (√(k+(√n))))/(Σ_(n=1) ^(k^2 −1) (√(k−(√n)))))=1+(√2)

$$\mathrm{is}\:\mathrm{this}\:\mathrm{true}\:\mathrm{and}\:\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{proof}? \\ $$$$\forall{k}\in\mathbb{N}\mid{k}>\mathrm{1} \\ $$$$\frac{\underset{{n}=\mathrm{1}} {\overset{{k}^{\mathrm{2}} −\mathrm{1}} {\sum}}\sqrt{{k}+\sqrt{{n}}}}{\underset{{n}=\mathrm{1}} {\overset{{k}^{\mathrm{2}} −\mathrm{1}} {\sum}}\sqrt{{k}−\sqrt{{n}}}}=\mathrm{1}+\sqrt{\mathrm{2}} \\ $$

Question Number 149341 Answers: 0 Comments: 3

Question Number 149349 Answers: 0 Comments: 0

Question Number 149339 Answers: 1 Comments: 0

...nice......mathematics... ln( 2) −Σ_(n=1 ) ^∞ ((ζ ( 2n+1 )−1)/(n + 1)) = ? ....m.n....

$$\:\:\:\:\:\:\:\:\:\:...{nice}......{mathematics}... \\ $$$$\:\:\:\:\:\:{ln}\left(\:\mathrm{2}\right)\:−\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\frac{\zeta\:\left(\:\mathrm{2}{n}+\mathrm{1}\:\right)−\mathrm{1}}{{n}\:+\:\mathrm{1}}\:=\:? \\ $$$$\:\:\:\:....{m}.{n}.... \\ $$

Question Number 149331 Answers: 1 Comments: 2

Question Number 149329 Answers: 1 Comments: 1

Question Number 149323 Answers: 2 Comments: 0

How many ordered pairs (a,b) with b < a < 100 , a,b ∈ N such that (a/b) both ((a+1)/(b+1)) are integers .

$${How}\:\:{many}\:\:{ordered}\:\:{pairs}\:\left({a},{b}\right)\:{with}\:\:{b}\:<\:{a}\:<\:\mathrm{100}\:\:,\:\:{a},{b}\:\in\:\mathbb{N}\:\:{such}\:\:{that}\:\:\frac{{a}}{{b}}\:\:{both}\:\:\:\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}\:\:{are}\:\:{integers}\:\:.\: \\ $$

Question Number 149322 Answers: 0 Comments: 5

((Σ_(n=1) ^(99) ((√(10 + (√n)))))/(Σ_(n=1) ^(99) ((√(10 - (√n)))))) = ?

$$\frac{\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\mathrm{99}} {\sum}}\left(\sqrt{\mathrm{10}\:+\:\sqrt{\boldsymbol{{n}}}}\right)}{\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\mathrm{99}} {\sum}}\left(\sqrt{\mathrm{10}\:-\:\sqrt{\boldsymbol{{n}}}}\right)}\:=\:? \\ $$

Question Number 149317 Answers: 2 Comments: 0

lim_(x→∞) (√(x^6 +5x^3 ))−x

$${lim}_{{x}\rightarrow\infty} \sqrt{{x}^{\mathrm{6}} +\mathrm{5}{x}^{\mathrm{3}} }−{x} \\ $$

Question Number 149309 Answers: 3 Comments: 0

if t=tanx what is the value of sinx and cosx ?

$${if}\:\:\:\:{t}={tanx}\:{what}\:{is}\:{the}\:{value}\:{of} \\ $$$${sinx}\:{and}\:{cosx}\:? \\ $$

Question Number 149296 Answers: 2 Comments: 0

if a ; b > 0 find [ (((a^2 + 4)(b^2 + 9))/(ab)) ]_(min) = ?

$${if}\:\:\:{a}\:;\:{b}\:>\:\mathrm{0} \\ $$$${find}\:\:\:\left[\:\frac{\left({a}^{\mathrm{2}} \:+\:\mathrm{4}\right)\left({b}^{\mathrm{2}} \:+\:\mathrm{9}\right)}{{ab}}\:\right]_{\boldsymbol{{min}}} =\:? \\ $$

Question Number 149292 Answers: 1 Comments: 0

if sin2𝛂 = - (1/3) find 3tg^2 (((3𝛑)/4) - 𝛂) = ?

$${if}\:\:\:{sin}\mathrm{2}\boldsymbol{\alpha}\:=\:-\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$${find}\:\:\:\mathrm{3}{tg}^{\mathrm{2}} \left(\frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{4}}\:-\:\boldsymbol{\alpha}\right)\:=\:? \\ $$

Question Number 149291 Answers: 2 Comments: 0

f(x) = 2sinx^2 −cos^2 x−2 f^′ (x) = ?

$${f}\left({x}\right)\:=\:\mathrm{2}{sinx}^{\mathrm{2}} −{cos}^{\mathrm{2}} {x}−\mathrm{2} \\ $$$${f}\:^{'} \left({x}\right)\:=\:? \\ $$

Question Number 152597 Answers: 0 Comments: 0

Question Number 149275 Answers: 0 Comments: 0

Question Number 149274 Answers: 1 Comments: 0

Question Number 149273 Answers: 3 Comments: 0

∫_0 ^(π/4) ((e^(tan x) sin^2 x)/(cos^4 x)) dx =?

$$\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{e}^{\mathrm{tan}\:\mathrm{x}} \:\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}{\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}\:\mathrm{dx}\:=? \\ $$

Question Number 149268 Answers: 4 Comments: 0

Question Number 149266 Answers: 1 Comments: 0

Solve for natural numbers: x^2 + y^2 +x + y = 3xy

$${Solve}\:{for}\:{natural}\:{numbers}: \\ $$$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+{x}\:+\:{y}\:=\:\mathrm{3}{xy} \\ $$

Question Number 149271 Answers: 2 Comments: 0

Question Number 149259 Answers: 1 Comments: 0

cos^(−1) ((√(sin (41°)sin (19°)+(3/4))) )=?

$$\:\mathrm{cos}^{−\mathrm{1}} \left(\sqrt{\mathrm{sin}\:\left(\mathrm{41}°\right)\mathrm{sin}\:\left(\mathrm{19}°\right)+\frac{\mathrm{3}}{\mathrm{4}}}\:\right)=? \\ $$

Question Number 149252 Answers: 4 Comments: 0

e^y =(sin x)(cos x) find (dy/dx)

$$ \\ $$$${e}^{{y}} =\left(\mathrm{sin}\:{x}\right)\left(\mathrm{cos}\:{x}\right) \\ $$$${find}\:\frac{{dy}}{{dx}} \\ $$

Question Number 149251 Answers: 2 Comments: 0

lim_(x→0^+ ) ((tan x (√(tan x))−sin x (√(sin x)))/(x^3 (√x))) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{tan}\:\mathrm{x}}−\mathrm{sin}\:\mathrm{x}\:\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\mathrm{3}} \:\sqrt{\mathrm{x}}}\:=? \\ $$$$ \\ $$

Question Number 149247 Answers: 0 Comments: 3

Question Number 149244 Answers: 0 Comments: 0

Let a,b,c be positive real numbers with sum 3. Prove that (1/a)+(1/b)+(1/c) ≥ (3/(2a^2 +bc))+(3/(2b^2 +ac))+(3/(2c^2 +ab))

$$\:\:{Let}\:{a},{b},{c}\:{be}\:{positive}\:{real}\:{numbers} \\ $$$${with}\:{sum}\:\mathrm{3}.\:{Prove}\:{that}\: \\ $$$$\:\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}{a}^{\mathrm{2}} +{bc}}+\frac{\mathrm{3}}{\mathrm{2}{b}^{\mathrm{2}} +{ac}}+\frac{\mathrm{3}}{\mathrm{2}{c}^{\mathrm{2}} +{ab}} \\ $$

Question Number 149241 Answers: 2 Comments: 0

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