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AllQuestion and Answers: Page 1704

Question Number 32296 Answers: 0 Comments: 0

let u_n = (((n+1)^α −n^α )/n^(α−1) ) with α>1 find lim_(n→∞) u_n .

$${let}\:\:{u}_{{n}} =\:\frac{\left({n}+\mathrm{1}\right)^{\alpha} \:\:−{n}^{\alpha} }{{n}^{\alpha−\mathrm{1}} }\:\:{with}\:\alpha>\mathrm{1}\:\:{find}\:{lim}_{{n}\rightarrow\infty} {u}_{{n}} \:\:. \\ $$

Question Number 32295 Answers: 0 Comments: 2

calculate Σ_(k=0) ^n (2k+1)(−1)^k .

$${calculate}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\left(\mathrm{2}{k}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{k}} \:\:. \\ $$

Question Number 32294 Answers: 0 Comments: 1

let u_1 =1 and u_2 =2 and u_n =u_(n−1) +u_(n−2) find u_n interms of n .

$${let}\:{u}_{\mathrm{1}} =\mathrm{1}\:{and}\:{u}_{\mathrm{2}} =\mathrm{2}\:{and}\:{u}_{{n}} ={u}_{{n}−\mathrm{1}} \:+{u}_{{n}−\mathrm{2}} \\ $$$${find}\:{u}_{{n}} \:{interms}\:{of}\:{n}\:. \\ $$

Question Number 32293 Answers: 1 Comments: 1

let u_0 = (√3) and u_(n+1) =(√(2+u_n ^2 )) calculate u_n interms of n.

$${let}\:{u}_{\mathrm{0}} =\:\sqrt{\mathrm{3}}\:\:{and}\:{u}_{{n}+\mathrm{1}} =\sqrt{\mathrm{2}+{u}_{{n}} ^{\mathrm{2}} } \\ $$$${calculate}\:{u}_{{n}} \:{interms}\:{of}\:{n}. \\ $$

Question Number 32291 Answers: 0 Comments: 0

let u_n = Σ_(k=1) ^n (1/(n+k)) prove that 0≤u_n ≤1 .

$${let}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{n}+{k}}\:{prove}\:{that}\:\mathrm{0}\leqslant{u}_{{n}} \leqslant\mathrm{1}\:. \\ $$

Question Number 32290 Answers: 0 Comments: 1

let give u_0 =1 and u_(n+1) =(√(1+(√u_n ))) prove that u_n is increasing .

$${let}\:{give}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:{u}_{{n}+\mathrm{1}} =\sqrt{\mathrm{1}+\sqrt{{u}_{{n}} }}\:\:{prove}\:{that}\:{u}_{{n}} \:{is} \\ $$$${increasing}\:. \\ $$

Question Number 32289 Answers: 0 Comments: 0

find lim_(x→0) ln ( ((e^(2x) −1)/x)) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:{ln}\:\left(\:\frac{{e}^{\mathrm{2}{x}} −\mathrm{1}}{{x}}\right)\:. \\ $$

Question Number 32288 Answers: 0 Comments: 0

study the function f(x)=(x^2 /(x+1)) e^(1/x) .

$${study}\:{the}\:{function}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{x}+\mathrm{1}}\:{e}^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 32287 Answers: 0 Comments: 0

1) for x>0 prove that (1/(x+1)) ≤ln(x+1)−lnx ≤ (1/x) 2) let u_n = Σ_(p=1) ^(kn) (1/p) find lim_(n→∞ ) u_n .

$$\left.\mathrm{1}\right)\:{for}\:{x}>\mathrm{0}\:{prove}\:{that}\:\frac{\mathrm{1}}{{x}+\mathrm{1}}\:\leqslant{ln}\left({x}+\mathrm{1}\right)−{lnx}\:\leqslant\:\frac{\mathrm{1}}{{x}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\sum_{{p}=\mathrm{1}} ^{{kn}} \:\frac{\mathrm{1}}{{p}}\:\:\:{find}\:{lim}_{{n}\rightarrow\infty\:} \:{u}_{{n}} . \\ $$

Question Number 32286 Answers: 0 Comments: 0

calculate lim_(x→0) (((e^x /(√(1+x))) −1−(x/2))/x^2 ) .

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\frac{{e}^{{x}} }{\sqrt{\mathrm{1}+{x}}}\:−\mathrm{1}−\frac{{x}}{\mathrm{2}}}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 32281 Answers: 0 Comments: 0

calculate lim_(x→∞) (√(x^2 +x+1)) −(√(x^2 −x+1)) .

$${calculate}\:{lim}_{{x}\rightarrow\infty} \sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:−\sqrt{{x}^{\mathrm{2}} \:−{x}+\mathrm{1}}\:\:. \\ $$

Question Number 32280 Answers: 0 Comments: 0

let u_n = e^(1/(n^2 +1)) −1 1) find a equivalent of u_n and lim_(n→∞) u_n 2) study the convergence of Σu_n .

$${let}\:{u}_{{n}} =\:{e}^{\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+\mathrm{1}}} \:\:−\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \:{and}\:{lim}_{{n}\rightarrow\infty} {u}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\:\Sigma{u}_{{n}} \:. \\ $$

Question Number 32279 Answers: 0 Comments: 0

find lim_(x→0) ((tan^2 x)/((1−cosx))) .((e^x −1)/x)

$${find}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{tan}^{\mathrm{2}} {x}}{\left(\mathrm{1}−{cosx}\right)}\:.\frac{{e}^{{x}} \:−\mathrm{1}}{{x}}\:\:\: \\ $$

Question Number 32282 Answers: 0 Comments: 0

let give f(x)=(√(1+x^2 )) find f^((n)) (o) .

$${let}\:{give}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\:\:{find}\:\:{f}^{\left({n}\right)} \left({o}\right)\:.\: \\ $$

Question Number 32278 Answers: 0 Comments: 0

calculate lim_(x→+∞) (x−1)cos((π/x)) .

$${calculate}\:{lim}_{{x}\rightarrow+\infty} \left({x}−\mathrm{1}\right){cos}\left(\frac{\pi}{{x}}\right)\:. \\ $$

Question Number 32277 Answers: 0 Comments: 0

for x_i ∈[0,1] prove that (1−x_1 )(1−x_2 )....(1−x_n ) ≥1−(x_1 +x_2 +.... +x_n ).

$${for}\:{x}_{{i}} \:\in\left[\mathrm{0},\mathrm{1}\right]\:{prove}\:{that} \\ $$$$\left(\mathrm{1}−{x}_{\mathrm{1}} \right)\left(\mathrm{1}−{x}_{\mathrm{2}} \right)....\left(\mathrm{1}−{x}_{{n}} \right)\:\geqslant\mathrm{1}−\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} \:+....\:+{x}_{{n}} \right). \\ $$

Question Number 32276 Answers: 0 Comments: 0

prove that Σ_(i=1) ^n (Π_(j=0) ^p (i+j))=((n(n+1)(n+2)...(n+p+1))/(p+2))

$${prove}\:{that}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\left(\prod_{{j}=\mathrm{0}} ^{{p}} \left({i}+{j}\right)\right)=\frac{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)...\left({n}+{p}+\mathrm{1}\right)}{{p}+\mathrm{2}} \\ $$

Question Number 32285 Answers: 0 Comments: 0

let f(x)= x^(n−1) ln(1+x) with n integr and n≥1 1) calculate f^((p)) (x) 2) find f^((n)) (x)

$${let}\:{f}\left({x}\right)=\:{x}^{{n}−\mathrm{1}} {ln}\left(\mathrm{1}+{x}\right)\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({p}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$

Question Number 32284 Answers: 0 Comments: 0

prove that ∀ x∈[0,(π/4)] x≤tanx≤2x 2) find α_n and β_n from R / α_n ≤ Σ_(k=2) ^n tan((π/(2n)))≤ β_n

$${prove}\:{that}\:\forall\:{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right]\:\:{x}\leqslant{tanx}\leqslant\mathrm{2}{x} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\alpha_{{n}} \:{and}\:\beta_{{n}} \:{from}\:{R}\:/\:\:\alpha_{{n}} \leqslant\:\sum_{{k}=\mathrm{2}} ^{{n}} \:{tan}\left(\frac{\pi}{\mathrm{2}{n}}\right)\leqslant\:\beta_{{n}} \\ $$

Question Number 32283 Answers: 1 Comments: 1

let give f(x)=x+2 −(√(x+1)) 1) find f^(−1) (x) inverse of f(x) 2) calculate (f^(−1) )^′ (x) .

$${let}\:{give}\:{f}\left({x}\right)={x}+\mathrm{2}\:−\sqrt{{x}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}^{−\mathrm{1}} \left({x}\right)\:{inverse}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\left({f}^{−\mathrm{1}} \right)^{'} \left({x}\right)\:. \\ $$

Question Number 32274 Answers: 0 Comments: 0

prove that Σ_(k=0) ^n (2^k /(x^2^k +1)) = (1/(x−1)) − (2^(n+1) /(x^(2^(n+1) ) −1)) .

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\frac{\mathrm{2}^{{k}} }{{x}^{\mathrm{2}^{{k}} } \:+\mathrm{1}}\:=\:\frac{\mathrm{1}}{{x}−\mathrm{1}}\:−\:\frac{\mathrm{2}^{{n}+\mathrm{1}} }{{x}^{\mathrm{2}^{{n}+\mathrm{1}} \:} −\mathrm{1}}\:\:. \\ $$

Question Number 32273 Answers: 1 Comments: 1

let put u_n =Σ_(k=1) ^n k(k!) 1) prove that u_n =(n+1)! −1 2) study the convergence of Σ_(n=1) ^∞ (1/u_n ) .

$${let}\:{put}\:{u}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}\left({k}!\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{u}_{{n}} =\left({n}+\mathrm{1}\right)!\:−\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{u}_{{n}} }\:. \\ $$

Question Number 32271 Answers: 0 Comments: 0

study the convergence of Σ_(n=1) ^∞ (((−1)^(n−1) )/n^(1/3) ) .

$${study}\:{the}\:{convergence}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\frac{\mathrm{1}}{\mathrm{3}}} }\:. \\ $$

Question Number 32269 Answers: 1 Comments: 0

find ∫ (x^3 /(√(1+x^2 ))) dx

$${find}\:\int\:\:\frac{{x}^{\mathrm{3}} }{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 32266 Answers: 0 Comments: 0

Question Number 32264 Answers: 0 Comments: 1

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