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Question Number 73247 Answers: 1 Comments: 0

Question Number 73243 Answers: 1 Comments: 0

prove that for z ∈C arctanz =(1/(2i))ln(((1+iz)/(1−iz)))

$p r o v e t h a t f o r z \in C a r c t a n z = \frac{1}{2 i} l n (\frac{1 + i z}{1 - i z})$

Question Number 73238 Answers: 1 Comments: 1

let 0<a<1 calculate ∫_0 ^∞ ((ln(t)t^(a−1) )/(1+t))dt and ∫_0 ^∞ ((ln^2 (t)t^(a−1) )/(1+t))dt

$l e t 0 < a < 1 c a l c u l a t e \int_{0}^{\infty} \frac{l n (t) t^{a - 1}}{1 + t} d t a n d \int_{0}^{\infty} \frac{{l n}^{2} (t) t^{a - 1}}{1 + t} d t$

Question Number 73235 Answers: 0 Comments: 0

Question Number 73261 Answers: 1 Comments: 1

calculate Σ_(n=1) ^∞ (n^4 +2n^2 −3)(x^n /(n!)) in case of convergence.

$c a l c u l a t e \sum_{n = 1}^{\infty} (n^{4} + 2 n^{2} - 3) \frac{x^{n}}{n!} i n c a s e o f c o n v e r g e n c e .$

Question Number 73231 Answers: 1 Comments: 1

find the sum of Σ_(n=0) ^∞ (n^2 −3n+1)e^(−n)

$f i n d t h e s u m o f \sum_{n = 0}^{\infty} (n^{2} - 3 n + 1) e^{- n}$

Question Number 73230 Answers: 0 Comments: 0

calculate A_n =∫_0 ^∞ ((1+x^n )/(2+x^(2n) ))dx and J_n =∫_0 ^∞ ((2+x^(3n) )/(5+x^(7n) ))dx with n integr natural not 0

$c a l c u l a t e A_{n} = \int_{0}^{\infty} \frac{1 + x^{n}}{2 + x^{2 n}} d x a n d J_{n} = \int_{0}^{\infty} \frac{2 + x^{3 n}}{5 + x^{7 n}} d x$ $w i t h n i n t e g r n a t u r a l n o t 0$

Question Number 73227 Answers: 1 Comments: 0

find the polynom T_n wich verify T_n (cosθ)=cos(nθ) ∀n integr ∀θ real 1) find T_0 ,T_1 and T_2 and prove that T_(n+2) =2x T_(n+1) −T_n 3) find deg(T_n ) and T_n (1) ,T_n (−1) 4) find T^′ (cosθ) for 0<θ<π and prove that (1−x^2 )T_n ′′−xT′_n +n^2 T_n =0 5) find roots of T_n and decompose T_n inside R[x] 6) find the value of Π_(k=0) ^(n−1) cos((((2k+1)π)/(2n)))

$f i n d t h e p o l y n o m T_{n} w i c h v e r i f y T_{n} (c o s θ) = c o s (n θ)$ $\forall n i n t e g r \forall θ r e a l$ $1) f i n d T_{0}, T_{1} a n d T_{2} a n d p r o v e t h a t$ $T_{n + 2} = 2 x T_{n + 1} - T_{n}$ $3) f i n d d e g (T_{n}) a n d T_{n} (1), T_{n} (- 1)$ $4) f i n d T^{^{'}} (c o s θ) f o r 0 < θ < π a n d p r o v e t h a t$ $(1 - x^{2}) T_{n}^{″} - {x T}_{n}^{'} + n^{2} T_{n} = 0$ $5) f i n d r o o t s o f T_{n} a n d d e c o m p o s e T_{n} i n s i d e R [x]$ $6) f i n d t h e v a l u e o f \prod_{k = 0}^{n - 1} c o s (\frac{(2 k + 1) π}{2 n})$