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Question Number 30483 Answers: 0 Comments: 0

we define the bernoulli polynomial B_n by b_0 =1 and ∀n∈ N^★ b_n ^′ =n b_(n−1) and ∫_0 ^1 b_n (t)dt=0 1) find b_n (1)−b_n (0) for n≥2 2) prove that b_n (x)=(−1)^n b_n (1−x)∀n∈N 3)calculate b_0 , b_1 ,b_2 ,b_3

$${we}\:{define}\:{the}\:{bernoulli}\:{polynomial}\:{B}_{{n}} \:{by} \\ $$$${b}_{\mathrm{0}} =\mathrm{1}\:{and}\:\forall{n}\in\:{N}^{\bigstar} \:\:\:{b}_{{n}} ^{'} ={n}\:{b}_{{n}−\mathrm{1}} \:\:{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {b}_{{n}} \left({t}\right){dt}=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{b}_{{n}} \left(\mathrm{1}\right)−{b}_{{n}} \left(\mathrm{0}\right)\:{for}\:{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:\:{prove}\:{that}\:{b}_{{n}} \left({x}\right)=\left(−\mathrm{1}\right)^{{n}} {b}_{{n}} \left(\mathrm{1}−{x}\right)\forall{n}\in{N} \\ $$$$\left.\mathrm{3}\right){calculate}\:{b}_{\mathrm{0}} ,\:{b}_{\mathrm{1}} ,{b}_{\mathrm{2}} \:,{b}_{\mathrm{3}} \\ $$

Question Number 30482 Answers: 0 Comments: 0

find S= Σ_(p≥1,q≥1 and D^ (p,q)=1) (1/(p^2 q^2 )) .

$${find}\:\:{S}=\:\sum_{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}\:{and}\:\hat {{D}}\left({p},{q}\right)=\mathrm{1}} \:\:\frac{\mathrm{1}}{{p}^{\mathrm{2}} {q}^{\mathrm{2}} }\:. \\ $$

Question Number 30481 Answers: 0 Comments: 0

find the value of s_1 = Σ_(p≥1,q≥1) (1/(p^2 q^2 )) and s_2 = Σ_(p≥1,q≥1 ,pdivide q) (1/(p^2 q^2 )) .

$${find}\:{the}\:{value}\:{of}\:\:{s}_{\mathrm{1}} =\:\sum_{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}} \:\:\frac{\mathrm{1}}{{p}^{\mathrm{2}} {q}^{\mathrm{2}} }\:\:\:{and} \\ $$$${s}_{\mathrm{2}} =\:\:\sum_{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}\:,{pdivide}\:{q}} \:\:\frac{\mathrm{1}}{{p}^{\mathrm{2}} {q}^{\mathrm{2}} }\:. \\ $$

Question Number 30480 Answers: 0 Comments: 0

let f(x)= Σ_(k=2) ^∞ (((−1)^k )/(x+k)) 1) find D_f 2)let put δ(x)= Σ_(n=1) ^∞ (((−1)^n )/n^x ) (Rieman alternate serie) find f(x) interms of δ(x).

$${let}\:{f}\left({x}\right)=\:\sum_{{k}=\mathrm{2}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{x}+{k}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right){let}\:{put}\:\delta\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{{x}} }\:\:\left({Rieman}\:{alternate}\:{serie}\right) \\ $$$${find}\:{f}\left({x}\right)\:{interms}\:{of}\:\delta\left({x}\right). \\ $$

Question Number 30479 Answers: 0 Comments: 0

integrate y^(′′) −2y^′ +y =t with y(0)=1

$${integrate}\:\:{y}^{''} \:−\mathrm{2}{y}^{'} \:+{y}\:={t}\:\:{with}\:{y}\left(\mathrm{0}\right)=\mathrm{1} \\ $$

Question Number 30478 Answers: 0 Comments: 0

let give l_i (x)= ∫_2 ^x (dt/(ln(t))) find a serie equal to l_i (x). x≥2.

$${let}\:{give}\:\:{l}_{{i}} \left({x}\right)=\:\int_{\mathrm{2}} ^{{x}} \:\:\:\frac{{dt}}{{ln}\left({t}\right)}\:{find}\:{a}\:{serie}\:{equal}\:{to}\:{l}_{{i}} \left({x}\right). \\ $$$${x}\geqslant\mathrm{2}. \\ $$

Question Number 30477 Answers: 0 Comments: 0

f function 2(×) derivable prove that L(f^′ )= pL(f) −f(o) and L(f^(′′) )=p^2 L(f)−pf(0)−f^′ (0) 2) let f(t)=tsin(wt) find L(f).

$${f}\:{function}\:\mathrm{2}\left(×\right)\:{derivable}\:{prove}\:{that} \\ $$$${L}\left({f}^{'} \right)=\:{pL}\left({f}\right)\:−{f}\left({o}\right)\:{and}\:{L}\left({f}^{''} \right)={p}^{\mathrm{2}} {L}\left({f}\right)−{pf}\left(\mathrm{0}\right)−{f}^{'} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{f}\left({t}\right)={tsin}\left({wt}\right)\:{find}\:{L}\left({f}\right). \\ $$

Question Number 30476 Answers: 1 Comments: 0

find L(cos^2 x) and L(sin^2 x) L is laplace transform.

$${find}\:{L}\left({cos}^{\mathrm{2}} {x}\right)\:{and}\:{L}\left({sin}^{\mathrm{2}} {x}\right)\:{L}\:{is}\:{laplace}\:{transform}. \\ $$

Question Number 30475 Answers: 0 Comments: 0

let give f_n (x)= ∫_(1/n) ^n ((sin(xt))/t) e^(−t) dt 1)find lim_(n→∞) f_n (x) 2)find another form of f_n (x) by calculating f_n ^′ (x).

$${let}\:{give}\:{f}_{{n}} \left({x}\right)=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\frac{{sin}\left({xt}\right)}{{t}}\:{e}^{−{t}} \:{dt} \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{n}\rightarrow\infty} {f}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{another}\:{form}\:{of}\:{f}_{{n}} \left({x}\right)\:{by}\:{calculating}\:{f}_{{n}} ^{'} \left({x}\right). \\ $$

Question Number 30462 Answers: 0 Comments: 0

Question Number 30456 Answers: 1 Comments: 0

proof that (a^2 /((a−b)(a−c)))+(b^2 /((b−c)(b−a)))+(c^2 /((c−a)(c−b)))= a+b+c

$${proof}\:{that} \\ $$$$\frac{{a}^{\mathrm{2}} }{\left({a}−{b}\right)\left({a}−{c}\right)}+\frac{{b}^{\mathrm{2}} }{\left({b}−{c}\right)\left({b}−{a}\right)}+\frac{{c}^{\mathrm{2}} }{\left({c}−{a}\right)\left({c}−{b}\right)}=\:{a}+{b}+{c} \\ $$

Question Number 30455 Answers: 0 Comments: 5

Question Number 30454 Answers: 0 Comments: 0

organic conversion:acetic acid from 2−methyle propane−2−ole

$$\mathrm{organic}\:\mathrm{conversion}:\mathrm{acetic}\:\mathrm{acid}\:\mathrm{from}\:\:\mathrm{2}−\mathrm{methyle}\:\mathrm{propane}−\mathrm{2}−\mathrm{ole} \\ $$

Question Number 30443 Answers: 0 Comments: 0

let w_n (x)=Σ_(k=1) ^n (x^k /k) find w_n (x) for ∣x∣<1 2) find lim_(n→∞) w_n (x).

$${let}\:\:{w}_{{n}} \left({x}\right)=\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{x}^{{k}} }{{k}}\:\:{find}\:{w}_{{n}} \left({x}\right)\:{for}\:\mid{x}\mid<\mathrm{1}\: \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} {w}_{{n}} \left({x}\right). \\ $$

Question Number 30442 Answers: 3 Comments: 2

prove that (1/e) ≤ ∫_0 ^1 e^(−(x−[x])^2 ) dx≤1.

$${prove}\:{that}\:\:\:\frac{\mathrm{1}}{{e}}\:\leqslant\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−\left({x}−\left[{x}\right]\right)^{\mathrm{2}} } \:{dx}\leqslant\mathrm{1}. \\ $$

Question Number 30441 Answers: 1 Comments: 0

prove that ∫_0 ^∞ e^(−[x]^2 ) = Σ_(n≥0) e^(−n^2 ) .

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left[{x}\right]^{\mathrm{2}} } =\:\sum_{{n}\geqslant\mathrm{0}} \:{e}^{−{n}^{\mathrm{2}} } . \\ $$

Question Number 30440 Answers: 0 Comments: 0

if (Σ_(n≥1) (((−1)^(n−1) )/n^x ))^2 = Σ_n c_(n ) (x) find c_n (x).

$${if}\:\:\:\left(\sum_{{n}\geqslant\mathrm{1}} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{{x}} }\right)^{\mathrm{2}} =\:\sum_{{n}} \:\:{c}_{{n}\:} \left({x}\right)\:\:{find}\:{c}_{{n}} \left({x}\right). \\ $$

Question Number 30439 Answers: 0 Comments: 0

let F(x)=Σ_(n=1) ^∞ (((−1)^(n−1) )/n^x ) calculate (dF/dx)(x).

$${let}\:{F}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{{x}} }\:\:{calculate}\:\frac{{dF}}{{dx}}\left({x}\right). \\ $$

Question Number 30438 Answers: 0 Comments: 0

let ξ(x)= Σ_(n=1) ^∞ (1/n^x ) prove that ξ(x)= γ +(1/(x−1)) +o(1).

$${let}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:{prove}\:{that}\: \\ $$$$\xi\left({x}\right)=\:\gamma\:+\frac{\mathrm{1}}{{x}−\mathrm{1}}\:+{o}\left(\mathrm{1}\right). \\ $$

Question Number 30435 Answers: 0 Comments: 0

let f_n (x)= Σ_(n=0) ^∞ e^(−nx) calculate ∫_1 ^e f_n (x)dx.

$${let}\:\:{f}_{{n}} \left({x}\right)=\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{e}^{−{nx}} \:\:{calculate}\:\:\int_{\mathrm{1}} ^{{e}} \:{f}_{{n}} \left({x}\right){dx}. \\ $$

Question Number 30434 Answers: 0 Comments: 0

find the nature of Σ_(n=0) ^∞ x^(n!) .

$${find}\:{the}\:{nature}\:{of}\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:{x}^{{n}!} \:. \\ $$

Question Number 30433 Answers: 0 Comments: 0

find the nature of the serie Σ_(n=1) ^∞ (n^(2n) /((n!)^2 )) .

$${find}\:{the}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{n}^{\mathrm{2}{n}} }{\left({n}!\right)^{\mathrm{2}} }\:. \\ $$

Question Number 30432 Answers: 0 Comments: 0

find lim_(n→∞ ) Σ_(k=1) ^n (k/n)e^(−(k^2 /n^2 )) .

$${find}\:\:{lim}_{{n}\rightarrow\infty\:} \sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{k}}{{n}}{e}^{−\frac{{k}^{\mathrm{2}} }{{n}^{\mathrm{2}} }} \:\:\:. \\ $$

Question Number 30431 Answers: 0 Comments: 0

let f(x)= Σ_(n=1) ^∞ (x^n /(1−x^n )) with x∈[0,1[ prove that f(x)∼_(x→1) ((ln(1−x))/(x−1)).

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{x}^{{n}} }{\mathrm{1}−{x}^{{n}} }\:\:{with}\:{x}\in\left[\mathrm{0},\mathrm{1}\left[\:\:{prove}\:{that}\right.\right. \\ $$$${f}\left({x}\right)\sim_{{x}\rightarrow\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}−{x}\right)}{{x}−\mathrm{1}}. \\ $$

Question Number 30429 Answers: 0 Comments: 1

What are the conditions for using L′hospital rule?

$${What}\:{are}\:{the}\:{conditions}\:{for}\:{using} \\ $$$${L}'{hospital}\:{rule}? \\ $$

Question Number 30428 Answers: 0 Comments: 0

integrate (1+t^2 )y^′ =ty +1+t^2 .

$${integrate}\:\:\left(\mathrm{1}+{t}^{\mathrm{2}} \right){y}^{'} ={ty}\:+\mathrm{1}+{t}^{\mathrm{2}} . \\ $$

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