Triple Series

Author

Eric W. Weisstein
January 3, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/TripleSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TripleSeries.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Convergent

(∑_(i=1)^∞)(∑_(j=1)^∞)(∑_(k=1)^∞)(1/(((i j k))^2))==((π^6)/216)

V7

Sum[(1/(((i j k))^2)),{i,∞},{j,∞},{k,∞}]

(∑_(i=1)^∞)(∑_(j=1)^∞)(∑_(k=1)^∞)(1/((2^i) (2^j) (2^k)))==1

V7

Sum[1/(2^i*2^j*2^k),{i,∞},{j,∞},{k,∞}]

Divergent

(∑_(i=1)^∞)(∑_(j=1)^∞)(∑_(k=1)^∞)(1/(((i+j+k))^2))==ComplexInfinity

This should give DirectedInfinity[1]

Sum[1/(i+j+k)^2,{i,∞},{j,∞},{k,∞}]

(∑_i^∞)(∑_j^∞)(∑_k^∞)(1/(i j k))==?

Sum[(1/((i j k))),{i,∞},{j,∞},{k,∞}]//Timing

Sum[(1./(i j k)),{i,10},{j,10},{k,10}]//Timing

Sum[(1./(i j k)),{i,100},{j,100},{k,100}]//Timing

Sum[(1./(i j k)),{i,500},{j,500},{k,500}]//Timing

NSum[(1/(i j k)),{i,∞},{j,∞},{k,∞}]//Timing