The-value-of-the-infinite-product-3-9-1-4-27-1-8-81-1-16-to-is-equal-to-

Question Number 12828 by 786786AM last updated on 03/May/17

The value of the infinite product

\sqrt{3} \cdot \sqrt[4]{9} \cdot \sqrt[8]{27} \cdot \sqrt[16]{81} \dots to \infty is equal to____.

Answered by prakash jain last updated on 04/May/17

3^{1 / 2} \cdot 3^{2 / 4} \cdot 3^{3 / 8} \dots = 3^{(\frac{1}{2} + \frac{2}{2^{2}} + \frac{3}{2^{3}} + \frac{4}{2^{4}} + \dots)} = 3^{S}

S = \frac{1}{2} + \frac{2}{2^{2}} + \frac{3}{2^{3}} + \frac{4}{2^{4}} + \dots (i)

\frac{S}{2} = \frac{1}{2^{2}} + \frac{2}{2^{3}} + \frac{3}{2^{4}} + \dots . (i i)

s u b t r a c t i n g (i i) f r o m (i)

\frac{S}{2} = \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \frac{1}{2^{4}} + \dots = \frac{1 / 2}{1 - 1 / 2} = 1

S = 2

3^{S} = 3^{2} = 9

Commented by mrW1 last updated on 04/May/17

v e r y n i c e t r i c k!

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