Integration Mania!

Calculus Level 4

$\large \int_{0}^{1} \int_{0}^{1}\int_{0}^{1} \int_{0}^{1}\int_{0}^{1} \int_{0}^{1} \frac{1}{1-xyzabc} \; dx \; dy \; dz \; da \; db\; dc$

Find the value of the closed form of the above integral to 4 decimal places.

The answer is 1.0173.

1 solution

Mark Hennings
Jun 1, 2017

Expanding the integrand as a series $\frac{1}{1 - xyzabc} \; = \; \sum_{n=0}^\infty x^ny^nz^na^nb^nc^n \hspace{2cm} 0 \le x,y,z,a,b,c < 1$ and applying the Monotone Convergence Theorem, the integral is equal to $\sum_{n=0}^\infty \frac{1}{(n+1)^6} \; = \; \zeta(6) \; = \; \tfrac{1}{945}\pi^6$ making the answer $\boxed{1.017343062}$ .

Are those all $\le 1$ ? Won't they diverge at $1$ .

Aditya Narayan Sharma - 4 years ago

The series will converge so long as $xyzabc < 1$ , so at all points except $x=y=z=a=b=c=1$ . It does not matter that the series diverges at this point, since the series of integrals converges. That is why I appealed to the MCT.

Mark Hennings - 4 years ago

Integration Mania!

The answer is 1.0173.

1 solution

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