Σ_(n=0) ^∞ (∫_0 ^1 (x^n /(1+x))dx)^2 =ln 2


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Question Number 141623 by qaz last updated on 21/May/21

$\underset{n = 0}{\sum^{\infty}} {(\int_{0}^{1} \frac{x^{n}}{1 + x} d x)}^{2} = l n 2$
	Answered by mindispower last updated on 21/May/21
	$(∫_0 ^1 (x^n /(1+x))dx)^2 =∫_0 ^1 ∫_0 ^1 (((xy)^n )/((1+x)(1+y)))dxdy ⇔Σ_(n≥0) ∫∫_0 ^1 (((xy)^n )/((1+x)(1+y)))dxdy =∫∫_0 ^1 (1/((1+x)(1+y)(1−xy)))dxdy S=∫_0 ^1 (1/((1+y)))∫_0 ^1 (dx/((1+x)(1−xy)))dy ∫_0 ^1 (dx/((1+x)(1−xy)))=∫_0 ^1 (1/((1+y))).(1/(1+x))+(y/(1+y)).(1/(1−xy))dx =((ln(2))/(1+y))−(1/(1+y))ln(1−y) S=∫_0 ^1 ((ln(2))/((1+y)^2 ))−(1/((1+y)^2 ))ln(1−y)dy =[−((ln(2))/(1+y))]_0 ^1 +lim_(x→1) ((ln(1−y))/(1+y))]_0 ^x +∫_0 ^x (1/((1+y)(1−y)))dy =((ln(2))/2)+lim_(x→1) ((ln(1−x))/(1+x))+(1/2)ln(((1+x)/(1−x))) =((ln(2))/2)+lim_(x→1) ((ln(1+x))/2)+ln(1−x){(1/(1+x))−(1/2)} =ln(2)+lim_(x→1) (((1−x)ln(1−x))/(2(1+x)))=ln(2) ⇔Σ_(n≥0) (∫_0 ^1 (x^n /(1+x))dx)^2 =ln(2)$
	${(\int_{0}^{1} \frac{x^{n}}{1 + x} d x)}^{2} = \int_{0}^{1} \int_{0}^{1} \frac{{(x y)}^{n}}{(1 + x) (1 + y)} d x d y$ $\Leftrightarrow \sum_{n ⩾ 0} \int \int_{0}^{1} \frac{{(x y)}^{n}}{(1 + x) (1 + y)} d x d y$ $= \int \int_{0}^{1} \frac{1}{(1 + x) (1 + y) (1 - x y)} d x d y$ $S = \int_{0}^{1} \frac{1}{(1 + y)} \int_{0}^{1} \frac{d x}{(1 + x) (1 - x y)} d y$ $\int_{0}^{1} \frac{d x}{(1 + x) (1 - x y)} = \int_{0}^{1} \frac{1}{(1 + y)} . \frac{1}{1 + x} + \frac{y}{1 + y} . \frac{1}{1 - x y} d x$ $= \frac{l n (2)}{1 + y} - \frac{1}{1 + y} l n (1 - y)$ $S = \int_{0}^{1} \frac{l n (2)}{{(1 + y)}^{2}} - \frac{1}{{(1 + y)}^{2}} l n (1 - y) d y$ ${= {[- \frac{l n (2)}{1 + y}]}_{0}^{1} + \lim_{x \to 1} \frac{l n (1 - y)}{1 + y}]}_{0}^{x} + \int_{0}^{x} \frac{1}{(1 + y) (1 - y)} d y$ $= \frac{l n (2)}{2} + \lim_{x \to 1} \frac{l n (1 - x)}{1 + x} + \frac{1}{2} l n (\frac{1 + x}{1 - x})$ $= \frac{l n (2)}{2} + \lim_{x \to 1} \frac{l n (1 + x)}{2} + l n (1 - x) {\frac{1}{1 + x} - \frac{1}{2}}$ $= l n (2) + \lim_{x \to 1} \frac{(1 - x) l n (1 - x)}{2 (1 + x)} = l n (2)$ $\Leftrightarrow \sum_{n ⩾ 0} {(\int_{0}^{1} \frac{x^{n}}{1 + x} d x)}^{2} = l n (2)$
	Commented by qaz last updated on 22/May/21

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