1)calculate A_n =∫_(1/n) ^1 ((ln(1+x^2 ))/(1+x^2 ))dx with n integr and n≥1 2) find lim_(n→+∞) A_n 3) study the convergence of Σ A


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Question Number 56329 by maxmathsup by imad last updated on 14/Mar/19

$1) c a l c u l a t e A_{n} = \int_{\frac{1}{n}}^{1} \frac{l n (1 + x^{2})}{1 + x^{2}} d x w i t h n i n t e g r a n d n ⩾ 1$ $2) f i n d {l i m}_{n \to + \infty} A_{n}$ $3) s t u d y t h e c o n v e r g e n c e o f Σ A_{n}$
Commented by maxmathsup by imad last updated on 17/Mar/19

$1) A_{n} =_{x = t a n θ} \int_{a r c t a n (\frac{1}{n})}^{\frac{π}{4}} \frac{l n (1 + {t a n}^{2} θ)}{1 + {t a n}^{2} θ} (1 + {t a n}^{2} θ) d θ$ $= \int_{a r c t a n (\frac{1}{n})}^{\frac{π}{4}} l n (\frac{1}{{c o s}^{2} θ}) d θ = - 2 \int_{a r c t a n (\frac{1}{n})}^{\frac{π}{4}} l n (c o s θ) d θ \Rightarrow$ ${l i m}_{n \to + \infty} A_{n} = - 2 \int_{0}^{\frac{π}{4}} l n (c o s θ) d θ l e t I = \int_{0}^{\frac{π}{4}} l n (c o s θ) d θ a n d$ $J = \int_{0}^{\frac{π}{4}} l n (s i n θ) d θ w e h a v e I + J = \int_{0}^{\frac{π}{4}} l n (c o s θ s i n θ) d θ$ $= \int_{0}^{\frac{π}{4}} l n (\frac{s i n (2 θ)}{2}) d θ = - \frac{π}{4} l n (2) + \int_{0}^{\frac{π}{4}} l n (s i n (2 θ)) d θ b u t$ $\int_{0}^{\frac{π}{4}} l n (s i n (2 θ) d θ =_{2 θ = t} \frac{1}{2} \int_{0}^{\frac{π}{2}} l n (s i n t) d t = \frac{1}{2} (- \frac{π}{2} l n (2)) = - \frac{π}{4} l n (2) \Rightarrow$ $I + J = - \frac{π}{2} l n (2)$ $I = \int_{0}^{\frac{π}{4}} l n (c o s θ) d θ =_{θ = t - \frac{π}{2}} \int_{\frac{π}{2}}^{\frac{3 π}{4}} l n (s i n θ) d θ = \int_{\frac{π}{2}}^{0} l n (s i n θ) d θ + \int_{0}^{\frac{3 π}{4}} l n (s i n θ) d θ$ $= \frac{π}{2} l n (2) + \int_{0}^{\frac{3 π}{4}} l n (s i n θ) d θ . . . . b e c o n t i n u e d . . . .$
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