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Integral of the fractional part of the reciprocal of x (video)

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http://faculty.missouristate.edu/l/lesreid/potw.html

Missouri State University’s Advanced Problem of February 2010

Vincent Pantaloni (Orléans, France)

Problem : Evaluate the following integral : Z 1

0

1 x

dx where ((x))denotes the fractional part of a numberx.

Solution :. . . .

∀x∈R, ((x)) =x−[x]. Where[x] is the integer part of x. The above integral is undefined in 0 so we have to evaluate :

n→+∞lim Z 1

n1

1 x

dx Letnbe an integer greater than 2 and consider :In=R1

1 n

1 x

dx.Then :

In= Z 1

n1

1 x−

1 x

dx

= Z 1

n1

1 x dx−

Z 1

n1

1 x

dx

= ln(n)− Z 1

1n

1 x

dx

= ln(n)−

n

X

k=2

Z k−11

1 k

1 x

dx

= ln(n)−

n

X

k=2

Z k−11

1 k

k−1 dx

= ln(n)−

n

X

k=2

1 k−1 −1

k

×(k−1)

= ln(n)−

n

X

k=2

1−k−1 k

= ln(n)−

n

X

k=2

1 k

= 1 + ln(n)−

n

X

k=1

1 k

We know that lim

n→+∞

Pn

k=1 1 k

−ln(n)

=γ. Whereγ≈0.5772156649is theEulerconstant. Hence :

Z 1

0

1 x

dx= 1−γ

http://prof.pantaloni.free.fr mail me

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