\(xf(x)\) \(= \sum_{p=0}^\infty \frac{x}{\sqrt{(2p+1)^2+x^2}} - \frac{x}{\sqrt{(2p+2)^2+x^2}}\)

\(= \sum_{p=0}^\infty \int_{t=2p+1}^{2p+2}\frac{xt}{(t^2+x^2)^{3/2}}\,d t\)

\(= \sum_{p=0}^\infty \int_{u=(2p+1)/x}^{(2p+2)/x}\frac{u}{(u^2+1)^{3/2}}\,d u\).

On a \(\int_{u=0}^\infty\frac u{(u^2+1)^{3/2}}\,d u = 1 = a + b\) avec :

\(a = \sum_{p=0}^\infty \int_{u=(2p)/x}^{(2p+1)/x}\frac{u}{(u^2+1)^{3/2}}\,d u\) et \(b = \sum_{p=0}^\infty \int_{u=(2p+1)/x}^{(2p+2)/x}\frac{u}{(u^2+1)^{3/2}}\,d u = xf(x)\).

\(h\) : \(u \mapsto\frac{u}{(u^2+1)^{3/2}}\) est croissante sur \([0,\sqrt{\frac12}]\) et décroissante sur \([\sqrt{\frac12},+\infty[\) donc \(|a-b| \le \frac{3\|h\|_\infty}x\), et \(xf(x)\to \frac12\) lorsque \(x\to+\infty\).

exo7 4558