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Question Number 140050 Answers: 3 Comments: 0

prove that:: φ :=lim_(n→∞) (n/( (√(2k)))) .(√(1−cos^k (((2π)/n)))) =π ................

$p r o v e t h a t ::$ $ϕ := {l i m}_{n \to \infty} \frac{n}{\sqrt{2 k}} . \sqrt{1 - {c o s}^{k} (\frac{2 π}{n})} = π$ $. . . . . . . . . . . . . . . .$

Question Number 140738 Answers: 0 Comments: 4

Question Number 140046 Answers: 2 Comments: 1

........... nice .......calculus(I) ........ Θ :=lim_( x→ (π/4)) ( tan(x) )^( tan(2x)) =? ...............................

$. . . . . . . . . . . n i c e . . . . . . . c a l c u l u s (I) . . . . . . . .$ $Θ := {l i m}_{x \to \frac{π}{4}} {(t a n (x))}^{t a n (2 x)} = ?$ $. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$

Question Number 140042 Answers: 1 Comments: 0

Question Number 140040 Answers: 3 Comments: 0

((45+29(√2)))^(1/3) + ((45−29(√2)))^(1/3) =?

$\sqrt[3]{45 + 29 \sqrt{2}} + \sqrt[3]{45 - 29 \sqrt{2}} = ?$

Question Number 140030 Answers: 1 Comments: 0

Question Number 140029 Answers: 1 Comments: 0

si p=ab and (a=1 ou b=1) then p is prime?

$s i p = a b a n d (a = 1 o u b = 1) t h e n p i s p r i m e ?$

Question Number 140028 Answers: 0 Comments: 0

Evaluate ::: P :=Π_(k=3) ^∞ (((k^3 +3k)^2 )/(k^6 −64))=? ..........................

$E v a l u a t e :::$ $P := \underset{k = 3}{\prod^{\infty}} \frac{{(k^{3} + 3 k)}^{2}}{k^{6} - 64} = ?$ $. . . . . . . . . . . . . . . . . . . . . . . . . .$

Question Number 140020 Answers: 0 Comments: 1

Question Number 140008 Answers: 1 Comments: 0

If in a number system 25×32=1163 find how many digits are there in the number system used.

$I f i n a n u m b e r s y s t e m$ $25 \times 32 = 1163$ $f i n d h o w m a n y d i g i t s a r e t h e r e$ $i n t h e n u m b e r s y s t e m u s e d .$

Question Number 140007 Answers: 2 Comments: 0

lim_(x→0) (1+(3/x))^x =?

$\lim_{x \to 0} {(1 + \frac{3}{x})}^{x} = ?$

Question Number 140002 Answers: 1 Comments: 0

Question Number 140000 Answers: 2 Comments: 2

lim_(x→0) (((1+x.2^x )/(1+x.3^x )))^(2/x^2 ) =?

$\lim_{x \to 0} {(\frac{1 + x . 2^{x}}{1 + x . 3^{x}})}^{\frac{2}{x^{2}}} = ?$

Question Number 139998 Answers: 1 Comments: 1

Question Number 139992 Answers: 0 Comments: 1

Question Number 139978 Answers: 1 Comments: 0

∫_( −1) ^( 5) ((1/2)x−2)∙5^(x^2 −4x−5) dx=?

$\underset{- 1}{\int^{5}} (\frac{1}{2} x - 2) \cdot 5^{x^{2} - 4 x - 5} d x = ?$

Question Number 139975 Answers: 2 Comments: 0

∫_0 ^∞ (cos(x)−(1/((1+x^2 ))))(dx/x)=−γ

$\int_{0}^{\infty} (c o s (x) - \frac{1}{(1 + x^{2})}) \frac{d x}{x} = - γ$

Question Number 139973 Answers: 3 Comments: 1

Question Number 139967 Answers: 1 Comments: 0

Question Number 139959 Answers: 1 Comments: 0

If W varies jointly as L and the square of R. Find the percentage change in W if L increases by 20% and R increases by 50%. If W=15 when H=3 and R=2(1/3). find W when H=1 and R=10. Find also W interms of H and R.

$I f W v a r i e s j o i n t l y a s L a n d t h e$ $s q u a r e o f R . F i n d t h e p e r c e n t a g e$ $c h a n g e i n W i f L i n c r e a s e s b y 20 %$ $a n d R i n c r e a s e s b y 50 % .$ $I f W = 15 w h e n H = 3 a n d R = 2 \frac{1}{3} . f i n d$ $W w h e n H = 1 a n d R = 10 .$ $F i n d a l s o W i n t e r m s o f H a n d R .$

Question Number 139957 Answers: 1 Comments: 4

Question Number 139954 Answers: 2 Comments: 0

Question Number 140141 Answers: 1 Comments: 0

......advanced calculus...... when ∣z∣<1 and:: Ω:=((sin(x))/(z^2 +2z cos(x)+1)) =Σ_(n=0) ^∞ a_n z^n are satisfied , then solve , a_n ... ..................

$. . . . . . a d v a n c e d c a l c u l u s . . . . . .$ $w h e n ∣ z ∣< 1 a n d ::$ $Ω := \frac{s i n (x)}{z^{2} + 2 z c o s (x) + 1} = \underset{n = 0}{\sum^{\infty}} a_{n} z^{n}$ $a r e s a t i s f i e d, t h e n s o l v e, a_{n} . . .$ $. . . . . . . . . . . . . . . . . .$

Question Number 139950 Answers: 1 Comments: 0

Question Number 139948 Answers: 3 Comments: 0

If x^2 +xy+y^2 =0 find :: ((x/(x+y)))^(2023) +((y/(x+y)))^(2023)

$I f x^{2} + x y + y^{2} = 0$ $f i n d :: {(\frac{x}{x + y})}^{2023} + {(\frac{y}{x + y})}^{2023}$

Question Number 139949 Answers: 0 Comments: 0

.........nice .... ∗ ....∗ .... ∗ ....calculus(I) 𝛗= lim_(x→0^+ ) (((cos((√x) ))^(1/3) )^(cot(x)) =?? solution......... sin(x) ≈ x (x → 0 ) 𝛗:= lim_(x→0^+ ) {(cos((√x) ))^(1/3) }^(cot(x)) =lim_(x→0^+ ) {(1−(x/2))^(cot(x)) }^(1/3) :=lim_(x→0^+ ) {(1−(x/2))^(1/(sin(x)≈x)) }^((cos(x))/3) =_(cos(x)→1 (x →0)) ^(⟨ lim_(x→0) (1−ax)^(1/x) =(1/e^( a) ) ⟩) e^((−1)/6) = (((1/e) ))^(1/6) ....

$. . . . . . . . . n i c e . . . . * . . . . * . . . . * . . . . c a l c u l u s (I)$ $ϕ = {l i m}_{x \to 0^{+}} {(\sqrt[3]{c o s (\sqrt{x}})}^{c o t (x)} = ? ?$ $s o l u t i o n . . . . . . . . .$ $s i n (x) \approx x (x \to 0)$ $ϕ := {l i m}_{x \to 0^{+}} {{(c o s (\sqrt{x}))}^{\frac{1}{3}}}^{c o t (x)} = {l i m}_{x \to 0^{+}} {{(1 - \frac{x}{2})}^{c o t (x)}}^{\frac{1}{3}}$ $:= {l i m}_{x \to 0^{+}} {{(1 - \frac{x}{2})}^{\frac{1}{s i n (x) \approx x}}}^{\frac{c o s (x)}{3}} \underset{c o s (x) \to 1 (x \to 0)}{\overset{⟨ {l i m}_{x \to 0} {(1 - a x)}^{\frac{1}{x}} = \frac{1}{e^{a}} ⟩}{=}} e^{\frac{- 1}{6}} = \sqrt[6]{\frac{1}{e}} . . . .$

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