Catalan's constant, is a mathematical constant, which is approximately equal to
It can be expressed as ,
where denotes the Dirichlet beta function.
Use the Maclaurin series expansion for the arctangent, and intergrate term by term:
Integrate by parts: Make the change of variable in above integral
Write the hyperbolic cosine in terms of exponentials, then expand in powers of using the geometric series formula. Integrate term by term. Then:
Let in and applying the double-angle formula for sine. Then:
Use the boundedly convergent Fourier series Integrate term by term, then:
Use , write sine in terms of cosine, and apply the well-known evaluation , then:
Add and , then:
Write the inverse hyperbolic sine as an integral whose integrand can be expressed in to powers of sine. And employing the reduction formula then: Replacing by immediately gives
Start with and write the arctangent in terms of the integral which defines it.
-plane. It moves in the following manner:Brilli the Bug has set out on a journey of infinite steps starting at the origin of the
- After each, th step, it turns counter-clockwise
- Each th step is of length where is given by for .
If the final displacement of brilli from the starting is given by , find .
Notation: denotes the Catalan's constant.