let A_n = ∫_0 ^n (((−1)^x )/(2[x] +1))dx 1) calculate A_n 2) find lim_(n→+∞) A


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Question Number 38804 by maxmathsup by imad last updated on 30/Jun/18

$l e t A_{n} = \int_{0}^{n} \frac{{(- 1)}^{x}}{2 [x] + 1} d x$ $1) c a l c u l a t e A_{n}$ $2) f i n d {l i m}_{n \to + \infty} A_{n}$
Commented by math khazana by abdo last updated on 30/Jun/18

$A_{n} = \int_{0}^{n} \frac{{(- 1)}^{[x]}}{2 [x] + 1} d x$
Commented by tanmay.chaudhury50@gmail.com last updated on 30/Jun/18

$\int_{0}^{1} \frac{{(- 1)}^{0}}{2 \times 0 + 2} d x + \int_{1}^{2} \frac{{(- 1)}^{1}}{2 \times 1 + 1} + \int_{2}^{3} \frac{{(- 1)}^{2}}{2 \times 2} + . . . +$
Commented by math khazana by abdo last updated on 01/Jul/18

$A_{n} = \sum_{k = 0}^{n - 1} \int_{k}^{k + 1} \frac{{(- 1)}^{k}}{2 k + 1} d x = \sum_{k = 0}^{n - 1} \frac{{(- 1)}^{k}}{2 k + 1}$ $2) {l i m}_{n \to + \infty} A_{n} = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{2 k + 1} = \frac{π}{4} .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 30/Jun/18

	$\int_{0}^{1} \frac{{(- 1)}^{0}}{2 \times 0 + 1} d x + \int_{1}^{2} \frac{{(- 1)}^{1}}{2 \times 1 + 1} d x + \int_{2}^{3} \frac{{(- 1)}^{2}}{2 \times 2 + 1} d x + . . .$ $+ \int_{n - 1}^{n} \frac{{(- 1)}^{n}}{2 (n - 1) + 1} d x$ $= \frac{1}{1} - \frac{1}{3} + \frac{1}{5} + . . . + \frac{{(- 1)}^{n}}{2 n - 1}$ $2) w e k n o w {t a n}^{- 1} x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + . . .$ $s o w h e n n \to \infty A_{n} \to {t a n}^{- 1} (1)$ $s o a n s i s$ $n \to \infty A_{n} \to \frac{Π}{4}$
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