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Question Number 46640 Answers: 0 Comments: 1

1≤n,m∈N. Prove that 3(m+n)+10ln (m!n!)≥6(√(mnH_m H_n )). (H_m =Σ_(i=1) ^m (1/i), H_n =Σ_(j=1) ^n (1/j))

$$\mathrm{1}\leqslant{n},{m}\in\mathbb{N}.\:{Prove}\:{that} \\ $$$$\mathrm{3}\left({m}+{n}\right)+\mathrm{10ln}\:\left({m}!{n}!\right)\geqslant\mathrm{6}\sqrt{{mnH}_{{m}} {H}_{{n}} }. \\ $$$$\left({H}_{{m}} =\underset{{i}=\mathrm{1}} {\overset{{m}} {\sum}}\frac{\mathrm{1}}{{i}},\:{H}_{{n}} =\underset{{j}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{j}}\right) \\ $$

Question Number 46639 Answers: 1 Comments: 2

Question Number 46641 Answers: 0 Comments: 1

Question Number 46637 Answers: 2 Comments: 3

Question Number 46636 Answers: 1 Comments: 0

tan θ=10tan60^°

$$\mathrm{tan}\:\theta=\mathrm{10tan60}^{°} \\ $$

Question Number 46631 Answers: 0 Comments: 1

Question Number 46629 Answers: 1 Comments: 0

An object is projected from a height of 80m above the ground with a velocity of 40m/s at an angle of 30 degree to the horizontal.What is the tume of flight?

$${An}\:{object}\:{is}\:{projected}\:{from}\:{a} \\ $$$${height}\:{of}\:\mathrm{80}{m}\:{above}\:{the}\:{ground} \\ $$$${with}\:{a}\:{velocity}\:{of}\:\mathrm{40}{m}/{s}\:{at}\:{an} \\ $$$${angle}\:{of}\:\mathrm{30}\:{degree}\:{to}\:{the} \\ $$$${horizontal}.{What}\:{is}\:{the}\:{tume}\:{of} \\ $$$${flight}? \\ $$

Question Number 46624 Answers: 1 Comments: 4

The value of k which minimizes F(k)= ∫_0 ^4 ∣x(4−x)−k∣dx = ?

$${The}\:{value}\:{of}\:{k}\:{which}\:{minimizes} \\ $$$${F}\left({k}\right)=\:\int_{\mathrm{0}} ^{\mathrm{4}} \mid{x}\left(\mathrm{4}−{x}\right)−{k}\mid{dx}\:=\:? \\ $$

Question Number 46617 Answers: 1 Comments: 1

calculate Σ_(n=1) ^∞ (1/(n(n+1)(n+2)(n+3)(n+4)(n+5)))

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)\left({n}+\mathrm{4}\right)\left({n}+\mathrm{5}\right)} \\ $$

Question Number 46612 Answers: 1 Comments: 4

1) calculate I_n = ∫_0 ^∞ x^n e^((1−i)x) dx with n integr natural and i^2 =−1 2) find ∫_0 ^∞ x^(4k+3) xsinx dx .

$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:{x}^{{n}} \:{e}^{\left(\mathrm{1}−{i}\right){x}} {dx}\:{with}\:{n}\:{integr}\:{natural}\:{and}\:{i}^{\mathrm{2}} =−\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\infty} \:{x}^{\mathrm{4}{k}+\mathrm{3}} \:{xsinx}\:{dx}\:. \\ $$

Question Number 46611 Answers: 2 Comments: 1

Question Number 46610 Answers: 0 Comments: 2

let f_n (x)=e^(−nx) −2e^(−2nx) with x from[0,+∞[ 1)calculate ∫_0 ^∞ f_n (x)dx and Σ_(n=0) ^∞ (∫_0 ^∞ f_n (x)dx) 2) find S(x)=Σ_(n=0) ^∞ f_n (x) and ∫_0 ^∞ S(x)dx

$${let}\:{f}_{{n}} \left({x}\right)={e}^{−{nx}} −\mathrm{2}{e}^{−\mathrm{2}{nx}} \:\:{with}\:{x}\:{from}\left[\mathrm{0},+\infty\left[\right.\right. \\ $$$$\left.\mathrm{1}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:{f}_{{n}} \left({x}\right){dx}\:\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\left(\int_{\mathrm{0}} ^{\infty} \:{f}_{{n}} \left({x}\right){dx}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:{f}_{{n}} \left({x}\right)\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:{S}\left({x}\right){dx} \\ $$

Question Number 46609 Answers: 0 Comments: 1

solve x y^(′′) −e^(−x) y^′ =x sinx

$${solve}\:\:\:\:{x}\:{y}^{''} \:−{e}^{−{x}} {y}^{'} \:\:\:={x}\:{sinx} \\ $$

Question Number 46608 Answers: 0 Comments: 0

let the d.e xy^(′′) +(x^2 −x)y^′ +2y =0 find a solution developpable at integr serie.

$${let}\:{the}\:{d}.{e}\:\:{xy}^{''} \:+\left({x}^{\mathrm{2}} −{x}\right){y}^{'} \:+\mathrm{2}{y}\:=\mathrm{0} \\ $$$${find}\:{a}\:{solution}\:{developpable}\:{at}\:{integr}\:{serie}. \\ $$$$ \\ $$

Question Number 46607 Answers: 0 Comments: 1

calculate Σ_((i,j)∈ N^2 ) ((i+j)/3^(i+j) )

$${calculate}\:\sum_{\left({i},{j}\right)\in\:{N}^{\mathrm{2}} } \:\:\:\:\:\frac{{i}+{j}}{\mathrm{3}^{{i}+{j}} } \\ $$

Question Number 46604 Answers: 0 Comments: 1

∫_0 ^(π/2) ((xcos x−sin x)/(x^2 +sin x)) dx = ?