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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 .

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

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 ) .

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

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