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Question Number 30177 Answers: 1 Comments: 0

let x∈R and u_n = Π_(k=0) ^n cos((x/2^k )) find a simple form of u_n .

$${let}\:{x}\in{R}\:\:{and}\:{u}_{{n}} =\:\prod_{{k}=\mathrm{0}} ^{{n}} \:{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)\:{find}\:{a}\:{simple}\:{form}\:{of} \\ $$$${u}_{{n}} . \\ $$

Question Number 30176 Answers: 0 Comments: 0

prove that v_n = Σ_(k=1) ^n (1/(2n+2k +1)) is convergente.

$${prove}\:{that}\:\:{v}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{2}{k}\:+\mathrm{1}}\:{is}\:{convergente}. \\ $$

Question Number 30175 Answers: 0 Comments: 2

prove that u_n = Σ_(k=1) ^n (1/(n+k)) is convergente .

$${prove}\:{that}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{n}+{k}}\:{is}\:{convergente}\:. \\ $$

Question Number 30174 Answers: 0 Comments: 0

let u_n = Σ_(k=1) ^n (1/k) 1. prove that ln(n+1)≤u_n ≤ln(n) +1 2. show that u_n _(n→∞) ∼ ln(n) .

$${let}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$$\mathrm{1}.\:{prove}\:{that}\:{ln}\left({n}+\mathrm{1}\right)\leqslant{u}_{{n}} \leqslant{ln}\left({n}\right)\:+\mathrm{1} \\ $$$$\mathrm{2}.\:{show}\:{that}\:{u}_{{n}} \:\:_{{n}\rightarrow\infty} \sim\:{ln}\left({n}\right)\:\:. \\ $$

Question Number 30193 Answers: 0 Comments: 0

let p_n (x)=−1 +Σ_(k=1) ^k x^k 1) prove that the equation p_n (x)=0 have only one solution x_n ∈[0,1] . 2) prove that (x_n ) is decreasing and minored by (1/2) 3) prove that lim_(n→∞) x_n =(1/2) .

$${let}\:{p}_{{n}} \left({x}\right)=−\mathrm{1}\:+\sum_{{k}=\mathrm{1}} ^{{k}} \:{x}^{{k}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{the}\:{equation}\:{p}_{{n}} \left({x}\right)=\mathrm{0}\:{have}\:{only}\:{one}\: \\ $$$${solution}\:{x}_{{n}} \in\left[\mathrm{0},\mathrm{1}\right]\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{decreasing}\:{and}\:{minored}\:{by}\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{n}\rightarrow\infty} \:{x}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\:. \\ $$

Question Number 30166 Answers: 0 Comments: 3

Question Number 30165 Answers: 0 Comments: 3

Question Number 30160 Answers: 0 Comments: 4

Question Number 30159 Answers: 1 Comments: 1

Question Number 30156 Answers: 1 Comments: 0

If x, y, z are in GP and x+3, y+3, z+3 are in HP, then

$$\mathrm{If}\:\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{in}\:\mathrm{GP}\:\mathrm{and}\:{x}+\mathrm{3},\:{y}+\mathrm{3},\:{z}+\mathrm{3}\:\mathrm{are} \\ $$$$\mathrm{in}\:\mathrm{HP},\:\mathrm{then} \\ $$

Question Number 30148 Answers: 2 Comments: 0

how to solve (√((c−(√(−c)))^2 ))+(√((c+6−(√(−c)))^2 )) 2<a<3, −4<b<−3, c=((a+b)/2)

$${how}\:{to}\:{solve}\:\sqrt{\left({c}−\sqrt{−{c}}\right)^{\mathrm{2}} }+\sqrt{\left({c}+\mathrm{6}−\sqrt{−{c}}\right)^{\mathrm{2}} } \\ $$$$\mathrm{2}<{a}<\mathrm{3},\:\:−\mathrm{4}<{b}<−\mathrm{3},\:\:{c}=\frac{{a}+{b}}{\mathrm{2}} \\ $$

Question Number 30129 Answers: 0 Comments: 1

Question Number 30123 Answers: 0 Comments: 3

find the convergence or divergence Σ_(n = 1) ^∞ (((n − 1)/n))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{or}\:\mathrm{divergence}\:\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\frac{\mathrm{n}\:−\:\mathrm{1}}{\mathrm{n}}\right) \\ $$

Question Number 30120 Answers: 2 Comments: 1

Question Number 30119 Answers: 2 Comments: 1

Let N= 2^(1224) −1. S= 2^(153) +2^(77) +1. T= 2^(408) −2^(204) +1. then which of the following statment is correct? a) S and T both divide N. b) only S divides N. c) only T divides N. d) Neither S nor T divides N.

$$\mathrm{Let}\:\:\mathrm{N}=\:\mathrm{2}^{\mathrm{1224}} \:−\mathrm{1}. \\ $$$$\mathrm{S}=\:\mathrm{2}^{\mathrm{153}} +\mathrm{2}^{\mathrm{77}} +\mathrm{1}. \\ $$$$\mathrm{T}=\:\mathrm{2}^{\mathrm{408}} −\mathrm{2}^{\mathrm{204}} +\mathrm{1}. \\ $$$$\mathrm{then}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{statment}\:\mathrm{is} \\ $$$$\:\mathrm{correct}? \\ $$$$\left.\mathrm{a}\right)\:\mathrm{S}\:\mathrm{and}\:\mathrm{T}\:\mathrm{both}\:\mathrm{divide}\:\mathrm{N}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{only}\:\mathrm{S}\:\mathrm{divides}\:\mathrm{N}. \\ $$$$\left.\mathrm{c}\right)\:\mathrm{only}\:\mathrm{T}\:\mathrm{divides}\:\mathrm{N}. \\ $$$$\left.\mathrm{d}\right)\:\mathrm{Neither}\:\mathrm{S}\:\mathrm{nor}\:\mathrm{T}\:\mathrm{divides}\:\mathrm{N}. \\ $$

Question Number 30106 Answers: 1 Comments: 0