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The value of the infinite product 6^(1/2...

The value of the infinite product `6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)xx...`

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`6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)xx...`
`=6^(1/2+2/4+3/8+4/16)`
Now, we will solve `(1/2+2/4+3/8+4/16+...)`.
Let `S = 1/2+2/4+3/8+4/16+...`
`S = 1/2+2/2^2+3/2^3+...->(1)`
`S/2 = 1/2^2+2/2^3+3/2^4+...->(2)`
Subtracting `(1) - (2)`,
`S/2 = 1/2+1/2^2+1/2^3+...`
...
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