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Prove that: 2^(1/4).4^(1/8), 8^(1/16). 1...

Prove that: `2^(1/4).4^(1/8), 8^(1/16). 16^(1/32)......... oo=2`.

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Given, `S=2^(1/4).4^(1/8), 8^(1/16). 16^(1/32)......... oo`
`S=2^(1/4+2/8+3/16+... oo)`
`S_1=1/4+2/8+3/16+... oo`` \ \ \ (1)`
Divide eq `(1)` by `2`
`S_2=1/8+2/16+3/32+... oo`` \ \ \ (2)`
Subtract `(2)` from `(1)`
`S_(3)=S_1-S_2=1/4+2/8+3/16+... oo-(1/8+2/16+3/32+... o o)`
`S_(3)=1/4+1/8+1/16+... oo`
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