The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$. Show that $\int _ { \lfloor x \rfloor } ^ { x } \frac { q ( u ) } { u } \mathrm {~d} u$ tends to 0 as $x$ tends to $+ \infty$, and deduce the convergence of the integral $\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { u } \mathrm {~d} u$, as well as the equality $$\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { u } \mathrm {~d} u = \frac { \ln ( 2 \pi ) } { 2 } - 1$$
The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $\int _ { \lfloor x \rfloor } ^ { x } \frac { q ( u ) } { u } \mathrm {~d} u$ tends to 0 as $x$ tends to $+ \infty$, and deduce the convergence of the integral $\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { u } \mathrm {~d} u$, as well as the equality
$$\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { u } \mathrm {~d} u = \frac { \ln ( 2 \pi ) } { 2 } - 1$$