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AllQuestion and Answers: Page 1789

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 .

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

Question Number 30427 Answers: 0 Comments: 6

Question Number 30426 Answers: 0 Comments: 0

find I_n = ∫_0 ^1 (dx/((1+x^2 )^n )) with n integr.

$${find}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}. \\ $$

Question Number 30424 Answers: 0 Comments: 0

find Σ_(n=0) ^∞ (x^n /(3n+2)) for ∣x∣<1 then find Σ_(n=0) ^∞ (((−1)^n )/((3n+2)2^n )) .

$${find}\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{{n}} }{\mathrm{3}{n}+\mathrm{2}}\:\:\:{for}\:\:\mid{x}\mid<\mathrm{1}\:\:{then}\:{find}\: \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{3}{n}+\mathrm{2}\right)\mathrm{2}^{{n}} }\:. \\ $$

Question Number 30423 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ ((sint)/t^α )dt . αfrom R.

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sint}}{{t}^{\alpha} }{dt}\:.\:\alpha{from}\:{R}. \\ $$

Question Number 30422 Answers: 0 Comments: 0

integrate the d.e. y^′ 2ty= sint

$${integrate}\:{the}\:{d}.{e}.\:{y}^{'} \:\mathrm{2}{ty}=\:{sint} \\ $$

Question Number 30421 Answers: 1 Comments: 0

integrate y^′ −2ty +ty^2 =0

$${integrate}\:{y}^{'} −\mathrm{2}{ty}\:+{ty}^{\mathrm{2}} =\mathrm{0} \\ $$

Question Number 30420 Answers: 1 Comments: 0

integrate y^(′′) = (1/2)(√(1+(y^′ )^2 )) .

$${integrate}\:{y}^{''} =\:\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{1}+\left({y}^{'} \right)^{\mathrm{2}} }\:\:\:\:\:. \\ $$

Question Number 30419 Answers: 1 Comments: 0

integrate (1+x^2 )y^′ +xy −2x=0 with cond.y(1)=0

$${integrate}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}^{'} \:+{xy}\:−\mathrm{2}{x}=\mathrm{0}\:{with}\:{cond}.{y}\left(\mathrm{1}\right)=\mathrm{0} \\ $$

Question Number 30418 Answers: 0 Comments: 0

integrate y^′ −2xy = sinx e^x^2 with y(0)=1.

$${integrate}\:{y}^{'} \:−\mathrm{2}{xy}\:=\:{sinx}\:{e}^{{x}^{\mathrm{2}} } \:{with}\:{y}\left(\mathrm{0}\right)=\mathrm{1}. \\ $$

Question Number 30417 Answers: 0 Comments: 0

integrate the d.e. (1+x^2 )y^′ −2x y = e^(−x^2 ) .

$${integrate}\:{the}\:{d}.{e}.\:\:\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}^{'} \:−\mathrm{2}{x}\:{y}\:=\:{e}^{−{x}^{\mathrm{2}} } . \\ $$

Question Number 30416 Answers: 0 Comments: 0

integrate the d.e. y^(′′) −4y =x +e^(2x) .

$${integrate}\:{the}\:{d}.{e}.\:\:\:{y}^{''} \:−\mathrm{4}{y}\:={x}\:+{e}^{\mathrm{2}{x}} . \\ $$

Question Number 30415 Answers: 0 Comments: 0

find the value of Σ_(k=0) ^n (1/(k+1)) C_n ^k .

$${find}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}+\mathrm{1}}\:{C}_{{n}} ^{{k}} \:\:. \\ $$

Question Number 30414 Answers: 0 Comments: 0

find the value of Σ_(p=0) ^n (−1)^(p ) (C_n ^p /(p+1)) .

$${find}\:\:{the}\:{value}\:{of}\:\:\sum_{{p}=\mathrm{0}} ^{{n}} \:\left(−\mathrm{1}\right)^{{p}\:\:} \:\frac{{C}_{{n}} ^{{p}} }{{p}+\mathrm{1}}\:. \\ $$

Question Number 30413 Answers: 0 Comments: 0

study the convergence of A(α)= ∫_0 ^∞ ((ln(t) arctant)/t^α )dt

$${study}\:{the}\:{convergence}\:{of}\:\:{A}\left(\alpha\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({t}\right)\:{arctant}}{{t}^{\alpha} }{dt} \\ $$

Question Number 30412 Answers: 0 Comments: 0

f is a function increazing(or decreazing)on ]0,1] prove that lim_(n→∞) (1/n)Σ_(q=1) ^n f((q/n))=∫_0 ^1 f(t)dt.

$$\left.{f}\left.\:{is}\:{a}\:{function}\:{increazing}\left({or}\:{decreazing}\right){on}\:\right]\mathrm{0},\mathrm{1}\right] \\ $$$${prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:\frac{\mathrm{1}}{{n}}\sum_{{q}=\mathrm{1}} ^{{n}} {f}\left(\frac{{q}}{{n}}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({t}\right){dt}. \\ $$$$ \\ $$

Question Number 30411 Answers: 0 Comments: 0

solve the d.e. y+x (y^′ )^3 =0

$${solve}\:{the}\:{d}.{e}.\:{y}+{x}\:\left({y}^{'} \right)^{\mathrm{3}} =\mathrm{0} \\ $$

Question Number 30409 Answers: 0 Comments: 0

find lim_(n→∞) Σ_(1≤i<j≤n) x^(i+j) .with ∣x∣<1 .

$${find}\:{lim}_{{n}\rightarrow\infty} \:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:{x}^{{i}+{j}} \:\:.{with}\:\mid{x}\mid<\mathrm{1}\:\:. \\ $$

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