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Question Number 34688 Answers: 0 Comments: 1

cslculate Σ_(n=2) ^∞ ln(1+(((−1)^n )/n))

$c s l c u l a t e \sum_{n = 2}^{\infty} l n (1 + \frac{{(- 1)}^{n}}{n})$

Question Number 34687 Answers: 0 Comments: 0

calculate Σ_(n=2) ^∞ ((1/(√(n−1))) + (1/(√(n+1))) −(2/(√n)))

$c a l c u l a t e \sum_{n = 2}^{\infty} (\frac{1}{\sqrt{n - 1}} + \frac{1}{\sqrt{n + 1}} - \frac{2}{\sqrt{n}})$

Question Number 34686 Answers: 0 Comments: 0

decompose F(x) = (((2n)!)/((x^2 −1)(x^2 −2)....(x^2 −n)))

$d e c o m p o s e F (x) = \frac{(2 n)!}{(x^{2} - 1) (x^{2} - 2) . . . . (x^{2} - n)}$

Question Number 34685 Answers: 0 Comments: 0

decompose the fraction F(x)= (1/((x+2)( x^n −1))) with n ∈ N^★

$d e c o m p o s e t h e f r a c t i o n$ $F (x) = \frac{1}{(x + 2) (x^{n} - 1)} w i t h n \in N^{★}$

Question Number 34684 Answers: 0 Comments: 0

let U_n = (π/4) −Σ_(k=0) ^n (((−1)^k )/(2k+1)) calcilate Σ_(n=0) ^∞ U_n

$l e t U_{n} = \frac{π}{4} - \sum_{k = 0}^{n} \frac{{(- 1)}^{k}}{2 k + 1}$ $c a l c i l a t e \sum_{n = 0}^{\infty} U_{n}$

Question Number 34683 Answers: 0 Comments: 0

find?the nature of Σ_(n=0) ^∞ sin{π(2+(√3) )^n }

$f i n d ? t h e n a t u r e o f \sum_{n = 0}^{\infty} s i n {π {(2 + \sqrt{3})}^{n}}$

Question Number 34682 Answers: 0 Comments: 0

calculate Σ_(n=0) ^∞ ln(cos((a/2^n )))

$c a l c u l a t e \sum_{n = 0}^{\infty} l n (c o s (\frac{a}{2^{n}}))$

Question Number 34681 Answers: 0 Comments: 0

calculate lim_(x→0) { ((1+tanx)/(1+thx))}^(1/(sinx)) .

$c a l c u l a t e {l i m}_{x \to 0} {\frac{1 + t a n x}{1 + t h x}}^{\frac{1}{s i n x}} .$

Question Number 34680 Answers: 0 Comments: 0

decompose inside C(x) the fraction F(x) = (x^2 /(x^4 −2x^2 cos(2a) +1)) .

$d e c o m p o s e i n s i d e C (x) t h e f r a c t i o n$ $F (x) = \frac{x^{2}}{x^{4} - 2 x^{2} c o s (2 a) + 1} .$

Question Number 34679 Answers: 0 Comments: 0

let f(x) = (x/(4x^2 −1)) 1) find f^((n)) (x) and f^((n)) (0) 2) developp f at ontegr serie .

$l e t f (x) = \frac{x}{4 x^{2} - 1}$ $1) f i n d f^{(n)} (x) a n d f^{(n)} (0)$ $2) d e v e l o p p f a t o n t e g r s e r i e .$

Question Number 34678 Answers: 0 Comments: 0

prove that ∀ n≥3 (√n) <^n (√(n!))

$p r o v e t h a t \forall n ⩾ 3 \sqrt{n} <^{n} \sqrt{n!}$

Question Number 34677 Answers: 0 Comments: 0

prove that Σ_(k=0) ^(n−1) [x +(k/n)] =[nx] ∀ n∈ ∈N^★

$p r o v e t h a t \sum_{k = 0}^{n - 1} [x + \frac{k}{n}] = [n x] \forall n \in \in N^{★}$

Question Number 34676 Answers: 0 Comments: 0

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

$p r o v e t h a t \sum_{k = 0}^{2 n - 1} \frac{{(- 1)}^{k}}{k + 1} = \sum_{k = n + 1}^{2 n} \frac{1}{k}$

Question Number 34675 Answers: 0 Comments: 0

provethat e = Σ_(k=0) ^n (1/(k!)) +∫_0 ^1 (((1−t)^n )/(n!)) e^t dt .

$p r o v e t h a t e = \sum_{k = 0}^{n} \frac{1}{k!} + \int_{0}^{1} \frac{{(1 - t)}^{n}}{n!} e^{t} d t .$

Question Number 34674 Answers: 0 Comments: 0

find ∫_0 ^π ((x sinx)/(1+cos^2 x)) dx

$f i n d \int_{0}^{π} \frac{x s i n x}{1 + {c o s}^{2} x} d x$

Question Number 34673 Answers: 0 Comments: 0

solve (((1+iz)/(1−iz)))^n = ((1+itanα)/(1−itanα)) with −(π/2)<α<(π/2)

$s o l v e {(\frac{1 + i z}{1 - i z})}^{n} = \frac{1 + i t a n α}{1 - i t a n α} w i t h - \frac{π}{2} < α < \frac{π}{2}$

Question Number 34672 Answers: 0 Comments: 0

prove that ∀n∈N ∣sin(nx)∣≤n∣sinx∣ .

$p r o v e t h a t \forall n \in N ∣ s i n (n x) ∣⩽ n ∣ s i n x ∣ .$

Question Number 34671 Answers: 0 Comments: 0

calculste Σ_(n=1) ^∞ arctan((2/n^2 )).

$c a l c u l s t e \sum_{n = 1}^{\infty} a r c t a n (\frac{2}{n^{2}}) .$

Question Number 34669 Answers: 0 Comments: 1

let P(x)=(1+x+ix^2 )^n −(1+x −ix^2 )^n 1) find the roots of P(x) 2) factorize inside C[x] P(x) 3) factorize indide R[x] P(x).

$l e t P (x) = {(1 + x + {i x}^{2})}^{n} - {(1 + x - {i x}^{2})}^{n}$ $1) f i n d t h e r o o t s o f P (x)$ $2) f a c t o r i z e i n s i d e C [x] P (x)$ $3) f a c t o r i z e i n d i d e R [x] P (x) .$

Question Number 34668 Answers: 0 Comments: 0

find the roots of?p(x) = x^(2n) −2x^n cos(nθ) +1 2)?factorize p(x)

$f i n d t h e r o o t s o f ? p (x) = x^{2 n} - 2 x^{n} c o s (n θ) + 1$ $2) ? f a c t o r i z e p (x)$

Question Number 34667 Answers: 0 Comments: 0

solve (x+1)^n = e^(2ina) then find the value of P_n = Π_(k=0) ^(n−1) sin(a +((kπ)/n))

$s o l v e {(x + 1)}^{n} = e^{2 i n a} t h e n f i n d t h e v a l u e o f$ $P_{n} = \prod_{k = 0}^{n - 1} s i n (a + \frac{k π}{n})$

Question Number 34666 Answers: 0 Comments: 0

simplify sin^2 ( ((arccosx)/2))

$s i m p l i f y {s i n}^{2} (\frac{a r c c o s x}{2})$

Question Number 34665 Answers: 0 Comments: 0

simplify sin (2arcsinx)

$s i m p l i f y s i n (2 a r c s i n x)$

Question Number 34664 Answers: 0 Comments: 0

simplify g(x)= arctan((1/(2x^2 ))) −arctan((x/(x+1))) +arctan(((x−1)/x))

$s i m p l i f y$ $g (x) = a r c t a n (\frac{1}{2 x^{2}}) - a r c t a n (\frac{x}{x + 1}) + a r c t a n (\frac{x - 1}{x})$

Question Number 34663 Answers: 0 Comments: 0

simplify f(x)=arcsin((√(1−x^2 ))) −arctan((√((1−x)/(1+x))))

$s i m p l i f y$ $f (x) = a r c s i n (\sqrt{1 - x^{2})} - a r c t a n (\sqrt{\frac{1 - x}{1 + x}})$

Question Number 34662 Answers: 0 Comments: 0

calculate I(a) =∫_(1/a) ^a ((ln(x))/(1+x^2 )) dx with a>0 2) calculate ∫_0 ^(+∞) ((ln(x))/(1+x^2 )) dx .

$c a l c u l a t e I (a) = \int_{\frac{1}{a}}^{a} \frac{l n (x)}{1 + x^{2}} d x w i t h a > 0$ $2) c a l c u l a t e \int_{0}^{+ \infty} \frac{l n (x)}{1 + x^{2}} d x .$

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