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Question Number 165215 Answers: 1 Comments: 0

Find: Σ_(n=1) ^∞ tan^(−1) ((1/n^2 ))

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{tan}^{−\mathrm{1}} \:\left(\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} }\right) \\ $$

Question Number 165210 Answers: 0 Comments: 1

Σ_(n=4) ^∞ 2 × ((3/4))^(n+3)

$$\underset{\boldsymbol{{n}}=\mathrm{4}} {\overset{\infty} {\sum}}\:\mathrm{2}\:×\:\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\boldsymbol{{n}}+\mathrm{3}} \\ $$

Question Number 165209 Answers: 0 Comments: 0

Find: Σ_(n=1) ^k ((n^2 sin^2 ((nπ)/2))/(n^4 π^4 + n^2 π^2 a + b)) = ?

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\sum}}\:\frac{\mathrm{n}^{\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}} \:\frac{\mathrm{n}\pi}{\mathrm{2}}}{\mathrm{n}^{\mathrm{4}} \:\pi^{\mathrm{4}} \:+\:\mathrm{n}^{\mathrm{2}} \:\pi^{\mathrm{2}} \:\mathrm{a}\:+\:\mathrm{b}}\:=\:? \\ $$

Question Number 165208 Answers: 1 Comments: 0

lim_(x→1_(y→2) ) x^2 +y^2 =?

$$\underset{{x}\rightarrow\underset{\mathrm{y}\rightarrow\mathrm{2}} {\mathrm{1}}} {\mathrm{lim}}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =? \\ $$

Question Number 165194 Answers: 2 Comments: 0

∫_0 ^( 2π) ln ( 1+ cos (x)).cos (nx )dx=?

$$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} {ln}\:\left(\:\mathrm{1}+\:{cos}\:\left({x}\right)\right).{cos}\:\left({nx}\:\right){dx}=? \\ $$

Question Number 165183 Answers: 0 Comments: 1

Question Number 165178 Answers: 2 Comments: 0

x∈R ⇒ ∣ log _2 ((x/2))∣^3 +∣log _2 (2x)∣^3 =28

$$\:\:{x}\in{R}\:\Rightarrow\:\mid\:\mathrm{log}\:_{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)\mid^{\mathrm{3}} +\mid\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{2}{x}\right)\mid^{\mathrm{3}} =\mathrm{28} \\ $$

Question Number 165170 Answers: 2 Comments: 3

Question Number 165168 Answers: 1 Comments: 0

Question Number 165164 Answers: 1 Comments: 0

Question Number 165162 Answers: 0 Comments: 0

f(x)=2x^2 +5x. Montrer que f est lipschitzienne sur R.

$${f}\left({x}\right)=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}. \\ $$$${Montrer}\:{que}\:{f}\:{est}\:{lipschitzienne} \\ $$$${sur}\:\mathbb{R}. \\ $$

Question Number 165161 Answers: 0 Comments: 0

f(x)=2x^2 +5x. Montrez que f est uniformement continue sur R.

$${f}\left({x}\right)=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}. \\ $$$${Montrez}\:{que}\:{f}\:{est}\:{uniformement}\:{continue}\: \\ $$$${sur}\:\mathbb{R}. \\ $$

Question Number 165160 Answers: 1 Comments: 0

f(x+f(x))=3f(x) and f(−1)=7 faind f(27)=?

$${f}\left({x}+{f}\left({x}\right)\right)=\mathrm{3}{f}\left({x}\right)\:\:\:{and}\:{f}\left(−\mathrm{1}\right)=\mathrm{7} \\ $$$${faind}\:\:{f}\left(\mathrm{27}\right)=? \\ $$

Question Number 165150 Answers: 1 Comments: 1

Question Number 165142 Answers: 0 Comments: 2

Question Number 165136 Answers: 3 Comments: 0

Question Number 165271 Answers: 1 Comments: 0

Let , f : [ 0 , 1 ] → R is a continuous function , prove that : lim_( n→ ∞) ∫_0 ^( 1) (( n f(x))/(1+ n^2 x^( 2) )) dx = (π/2) f (0 ) −−− proof −−− S_( n) = [∫_(0 ) ^( (1/( (√n)))) (( n. f(x))/(1 + n^( 2) x^( 2) )) dx =Ω_( n) ]+[ ∫_(1/( (√n))) ^( 1) ((n.f (x))/(1 + n^( 2) x^( 2) )) dx = Φ_( n) ] Ω_( n) =_(∃ t_( n) ∈ ( 0 , (1/( (√n) )) )) ^(MeanValueTheorem( first)) f (t_( n) )∫_(0 ) ^( (1/( (√n)))) (( n)/(1 + n^( 2) x^( 2) ))dx = f ( t_( n) ) ( tan^( −1) ( (√n) )) lim_( n→∞) (Ω_( n) ) = (π/2) f (lim_( n→∞) ( t_( n) ) ) = (π/2) f (0 ) Φ_( n) = ∫_(1/( (√n))) ^( 1) (( n. f(x) )/(1 + n^( 2) x^( 2) )) dx ⇒_(∃ M >0) ^(f is bounded) ∣ Φ_( n) ∣ ≤ M.∫_(1/( (√n))) ^( 1) (n/(1+ n^( 2) x^( 2) )) dx ⇒ ∣ Φ_( n) ∣ ≤ M . ( tan^( −1) ( n )− tan^( −1) ( (√n) )) lim_( n→ ∞) ∣ Φ_( n) ∣ = 0 ⇒ lim_( n→∞) Φ_( n) =0 ∴ lim_( n→ ∞) ( S_( n) ) = (π/2) f (0 ) ■ m.n

$$ \\ $$$$\:\:\:\:\:\mathrm{L}{et}\:,\:\:\:{f}\::\:\left[\:\mathrm{0}\:,\:\mathrm{1}\:\right]\:\rightarrow\:\mathbb{R}\:\:{is}\:{a}\:{continuous}\: \\ $$$$\:\:\:\:{function}\:,\:{prove}\:{that}\::\:\:\: \\ $$$$\:\:\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\:\infty} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{n}\:{f}\left({x}\right)}{\mathrm{1}+\:{n}^{\mathrm{2}} \:{x}^{\:\mathrm{2}} }\:{dx}\:=\:\frac{\pi}{\mathrm{2}}\:{f}\:\left(\mathrm{0}\:\right) \\ $$$$\:\:\:\:\:−−−\:{proof}\:−−− \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{S}_{\:{n}} \:=\:\left[\int_{\mathrm{0}\:} ^{\:\frac{\mathrm{1}}{\:\sqrt{{n}}}} \:\frac{\:{n}.\:{f}\left({x}\right)}{\mathrm{1}\:+\:{n}^{\:\mathrm{2}} {x}^{\:\mathrm{2}} }\:{dx}\:=\Omega_{\:{n}} \right]+\left[\:\int_{\frac{\mathrm{1}}{\:\sqrt{{n}}}} ^{\:\mathrm{1}} \:\frac{{n}.{f}\:\left({x}\right)}{\mathrm{1}\:+\:{n}^{\:\mathrm{2}} {x}^{\:\mathrm{2}} }\:{dx}\:=\:\Phi_{\:{n}} \:\right] \\ $$$$\:\:\:\:\:\:\Omega_{\:{n}} \:\underset{\exists\:{t}_{\:{n}} \in\:\left(\:\mathrm{0}\:,\:\frac{\mathrm{1}}{\:\sqrt{{n}}\:}\:\right)} {\overset{\mathrm{M}{ean}\mathrm{V}{alue}\mathrm{T}{heorem}\left(\:{first}\right)} {=}}\:\:{f}\:\left({t}_{\:{n}} \:\right)\int_{\mathrm{0}\:} ^{\:\frac{\mathrm{1}}{\:\sqrt{{n}}}} \:\frac{\:{n}}{\mathrm{1}\:+\:{n}^{\:\mathrm{2}} {x}^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:{f}\:\left(\:{t}_{\:{n}} \right)\:\left(\:{tan}^{\:−\mathrm{1}} \left(\:\sqrt{{n}}\:\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\infty} \:\left(\Omega_{\:{n}} \:\right)\:\:=\:\frac{\pi}{\mathrm{2}}\:{f}\:\left({lim}_{\:{n}\rightarrow\infty} \left(\:{t}_{\:{n}} \right)\:\right)\:=\:\frac{\pi}{\mathrm{2}}\:{f}\:\left(\mathrm{0}\:\right)\: \\ $$$$\:\:\:\:\:\:\:\:\Phi_{\:{n}} =\:\int_{\frac{\mathrm{1}}{\:\sqrt{{n}}}} ^{\:\mathrm{1}} \frac{\:{n}.\:{f}\left({x}\right)\:}{\mathrm{1}\:+\:{n}^{\:\mathrm{2}} {x}^{\:\mathrm{2}} }\:{dx}\:\:\underset{\exists\:\mathrm{M}\:>\mathrm{0}} {\overset{{f}\:\:{is}\:{bounded}} {\Rightarrow}}\:\mid\:\Phi_{\:{n}} \:\mid\:\leqslant\:\mathrm{M}.\int_{\frac{\mathrm{1}}{\:\sqrt{{n}}}} ^{\:\mathrm{1}} \frac{{n}}{\mathrm{1}+\:{n}^{\:\mathrm{2}} {x}^{\:\mathrm{2}} }\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\mid\:\Phi_{\:{n}} \mid\:\leqslant\:\mathrm{M}\:.\:\left(\:{tan}^{\:−\mathrm{1}} \left(\:{n}\:\right)−\:{tan}^{\:−\mathrm{1}} \left(\:\sqrt{{n}}\:\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\:\infty} \:\mid\:\Phi_{\:{n}} \mid\:=\:\mathrm{0}\:\:\Rightarrow\:{lim}_{\:{n}\rightarrow\infty} \:\Phi_{\:{n}} \:=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\:{lim}_{\:{n}\rightarrow\:\infty} \:\left(\:\:\mathrm{S}_{\:{n}} \:\right)\:=\:\frac{\pi}{\mathrm{2}}\:{f}\:\left(\mathrm{0}\:\right)\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$\:\:\:\:\: \\ $$

Question Number 165270 Answers: 2 Comments: 0