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Question Number 162478 Answers: 1 Comments: 0

Calculate: Σ_(k=1) ^∞ ((H_k 2^(-k) )/(k + 1)) where H_k is the k-th harmonic number

$$\mathrm{Calculate}:\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{H}_{\boldsymbol{\mathrm{k}}} \:\mathrm{2}^{-\boldsymbol{\mathrm{k}}} }{\mathrm{k}\:+\:\mathrm{1}} \\ $$$$\mathrm{where}\:\mathrm{H}_{\boldsymbol{\mathrm{k}}} \:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{k}}-\mathrm{th}\:\mathrm{harmonic}\:\mathrm{number} \\ $$

Question Number 162473 Answers: 1 Comments: 0

Question Number 162471 Answers: 2 Comments: 0

[reposted] find ∫_( 0) ^( (𝛑/2)) sin^8 (x)dx + ∫_( 0) ^( 1) sin^(-1) ((x)^(1/8) ) dx=?

$$\left[{reposted}\right] \\ $$$${find}\:\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\mathrm{sin}^{\mathrm{8}} \left(\mathrm{x}\right){dx}\:+\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{sin}^{-\mathrm{1}} \left(\sqrt[{\mathrm{8}}]{\mathrm{x}}\right)\:{dx}=? \\ $$

Question Number 162429 Answers: 0 Comments: 1

put the digits 0,1,2,3,4,5,6,7,8,9,in place of the letters in order to perform the edditon

$${put}\:{the}\:{digits}\:\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9},{in}\:{place}\:{of}\:{the}\:{letters}\:{in}\:{order}\:{to}\:{perform}\:{the}\:{edditon} \\ $$

Question Number 162424 Answers: 1 Comments: 0

calculate Ω = Σ_(n=1) ^∞ (( (−1)^( n) n)/(3^( n) (2n −1 ))) =? − Inspired from Sir Ghaderi′s post−

$$ \\ $$$$\:\:\:\:\:{calculate}\: \\ $$$$ \\ $$$$\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\:\left(−\mathrm{1}\right)^{\:{n}} {n}}{\mathrm{3}^{\:{n}} \:\left(\mathrm{2}{n}\:−\mathrm{1}\:\right)}\:=?\:\:\:\: \\ $$$$\:\:\:\:−\:\mathrm{I}{nspired}\:{from}\:{Sir}\:\mathrm{G}{haderi}'{s}\:{post}− \\ $$

Question Number 162533 Answers: 5 Comments: 0

Question Number 162416 Answers: 1 Comments: 1

Prove that: ∫_( 0) ^( (𝛑/4)) ((4 ln (cotx))/(cos(2x + 2022𝛑))) dx = 3𝛇(2)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {\int}}\:\frac{\mathrm{4}\:\mathrm{ln}\:\left(\mathrm{cot}\boldsymbol{\mathrm{x}}\right)}{\mathrm{cos}\left(\mathrm{2x}\:+\:\mathrm{2022}\boldsymbol{\pi}\right)}\:\mathrm{dx}\:=\:\mathrm{3}\boldsymbol{\zeta}\left(\mathrm{2}\right) \\ $$

Question Number 162417 Answers: 0 Comments: 4

Prove ∫_( 0) ^( (𝛑/2)) sin^8 (x) + ∫_( 0) ^( 1) sin^(-1) ((x)^(1/8) ) ≥ (π/2)

$$\mathrm{Prove} \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\mathrm{sin}^{\mathrm{8}} \left(\mathrm{x}\right)\:+\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{sin}^{-\mathrm{1}} \:\left(\sqrt[{\mathrm{8}}]{\mathrm{x}}\right)\:\geqslant\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 162414 Answers: 1 Comments: 0

Prove the Identity for any (a,n) in Real Number (1 + a)∙a^([n]) = a ∙ a^(2[(n/2)]) + a^(2[((n+1)/2)]) [∗] Greatest Integer Function

$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{Identity}\:\mathrm{for}\:\mathrm{any}\:\left(\mathrm{a},\mathrm{n}\right)\:\mathrm{in}\:\mathrm{Real}\:\mathrm{Number} \\ $$$$\left(\mathrm{1}\:+\:\mathrm{a}\right)\centerdot\mathrm{a}^{\left[\boldsymbol{\mathrm{n}}\right]} \:=\:\mathrm{a}\:\centerdot\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}}{\mathrm{2}}\right]} \:+\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}+\mathrm{1}}{\mathrm{2}}\right]} \\ $$$$\left[\ast\right]\:\mathrm{Greatest}\:\mathrm{Integer}\:\mathrm{Function} \\ $$

Question Number 162411 Answers: 0 Comments: 0

Prove the identity for any ′n′ in Real number [(n/2)] ∙ [((n + 1)/2)] = (1/4)([n]^2 + 2[(n/2)] - [n]) [∗] Greatest Integer Function

$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{identity}\:\mathrm{for}\:\mathrm{any}\:'\boldsymbol{\mathrm{n}}'\:\mathrm{in}\:\mathrm{Real}\:\mathrm{number} \\ $$$$\left[\frac{\mathrm{n}}{\mathrm{2}}\right]\:\centerdot\:\left[\frac{\mathrm{n}\:+\:\mathrm{1}}{\mathrm{2}}\right]\:=\:\frac{\mathrm{1}}{\mathrm{4}}\left(\left[\mathrm{n}\right]^{\mathrm{2}} \:+\:\mathrm{2}\left[\frac{\mathrm{n}}{\mathrm{2}}\right]\:-\:\left[\mathrm{n}\right]\right) \\ $$$$\left[\ast\right]\:\mathrm{Greatest}\:\mathrm{Integer}\:\mathrm{Function} \\ $$

Question Number 162410 Answers: 1 Comments: 0

∫(dx/((a−cosx)^2 )) a>1

$$\int\frac{{dx}}{\left({a}−{cosx}\right)^{\mathrm{2}} }\:\:\:{a}>\mathrm{1} \\ $$

Question Number 162510 Answers: 1 Comments: 0

lim_(x→0) ((7tan x−tan 7x)/(3x)) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{7tan}\:{x}−\mathrm{tan}\:\mathrm{7}{x}}{\mathrm{3}{x}}\:=? \\ $$

Question Number 162509 Answers: 0 Comments: 0

find Σ_(n=1) ^∞ (((−1)^n )/(n^3 (2n+1)^4 ))

$$\mathrm{find}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}^{\mathrm{3}} \left(\mathrm{2n}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 162399 Answers: 1 Comments: 1

Let m & n be two positive numbers greater than 1 . If lim_(p→0) ((e^(cos (p^n )) −e)/p^m ) = (1/2)e then (n/m)=?

$$\:\:{Let}\:{m}\:\&\:{n}\:{be}\:{two}\:{positive}\:{numbers}\: \\ $$$$\:{greater}\:{than}\:\mathrm{1}\:.\:{If}\:\underset{{p}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{\mathrm{cos}\:\left({p}^{{n}} \right)} −{e}}{{p}^{{m}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}{e}\: \\ $$$$\:{then}\:\frac{{n}}{{m}}=? \\ $$

Question Number 162398 Answers: 1 Comments: 0

lim_(x→0) ((∫_0 ^1 (arctan (t+sin x)−arctan t)dt)/(arctan x))=?

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{arctan}\:\left(\mathrm{t}+\mathrm{sin}\:\mathrm{x}\right)−\mathrm{arctan}\:\mathrm{t}\right)\mathrm{dt}}{\mathrm{arctan}\:\mathrm{x}}=? \\ $$

Question Number 162396 Answers: 1 Comments: 1

Question Number 162395 Answers: 0 Comments: 1

Question Number 162390 Answers: 0 Comments: 0

Question Number 162382 Answers: 2 Comments: 0

Question Number 162377 Answers: 1 Comments: 2

prove that ψ′′ ((1/4) )= −2π^( 3) − 56 ζ (3 )

$$ \\ $$$$\:\:{prove}\:\:{that} \\ $$$$ \\ $$$$\:\:\:\:\:\:\psi''\:\left(\frac{\mathrm{1}}{\mathrm{4}}\:\right)=\:−\mathrm{2}\pi^{\:\mathrm{3}} −\:\mathrm{56}\:\zeta\:\left(\mathrm{3}\:\right) \\ $$$$ \\ $$

Question Number 162374 Answers: 0 Comments: 2

Question Number 162371 Answers: 2 Comments: 0

If x ∈R the maximum value of ((3x^2 +9x+17)/(3x^2 +9x+7)) is ...

$$\:\:{If}\:{x}\:\in\mathbb{R}\:{the}\:{maximum}\:{value}\: \\ $$$$\:{of}\:\frac{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{9}{x}+\mathrm{17}}{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{9}{x}+\mathrm{7}}\:{is}\:... \\ $$

Question Number 162368 Answers: 1 Comments: 2

Question Number 162367 Answers: 1 Comments: 0

Let x_1 ,x_2 ,x_3 be the roots of the equation x^3 +3x+5=0 . Then the value of expression (x_1 +(1/x_1 ))(x_2 +(1/x_2 ))(x_3 +(1/x_3 )) is equal to

$$\:\:{Let}\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,{x}_{\mathrm{3}} \:{be}\:{the}\:{roots}\:{of}\:{the}\: \\ $$$${equation}\:{x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{5}=\mathrm{0}\:.\:{Then}\:{the} \\ $$$${value}\:{of}\:{expression}\:\left({x}_{\mathrm{1}} +\frac{\mathrm{1}}{{x}_{\mathrm{1}} }\right)\left({x}_{\mathrm{2}} +\frac{\mathrm{1}}{{x}_{\mathrm{2}} }\right)\left({x}_{\mathrm{3}} +\frac{\mathrm{1}}{{x}_{\mathrm{3}} }\right)\:{is} \\ $$$$\:{equal}\:{to} \\ $$

Question Number 162366 Answers: 1 Comments: 0

Given that the solution set of the quadratic inequality ax^2 +bx+c >0 is (2,3). Then the solution set of the inequality cx^2 +bx+a <0 will be

$$\:{Given}\:{that}\:{the}\:{solution}\:{set}\:{of}\:{the}\: \\ $$$$\:{quadratic}\:{inequality}\:{ax}^{\mathrm{2}} +{bx}+{c}\:>\mathrm{0} \\ $$$$\:{is}\:\left(\mathrm{2},\mathrm{3}\right).\:{Then}\:{the}\:{solution}\:{set}\: \\ $$$$\:{of}\:{the}\:{inequality}\:{cx}^{\mathrm{2}} +{bx}+{a}\:<\mathrm{0}\: \\ $$$$\:{will}\:{be}\: \\ $$

Question Number 162365 Answers: 0 Comments: 0

∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ln^2 (x+y+z)dxdydz=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{ln}^{\mathrm{2}} \left({x}+{y}+{z}\right){dxdydz}=? \\ $$

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