We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
109
1
avatar+187 

Determine the value of the infinite product \((2^\frac{1}{3})(4^\frac{1}{9})(8^\frac{1}{27}) ...\) 

 Aug 9, 2019
 #1
avatar+6045 
+1

\(\text{It's probably easiest to derive the log of this product and the take the exponential of the log.}\\ p = \prod \limits_{k=1}^\infty \left(2^k\right)^{\frac{1}{3^k}}= \prod \limits_{k=1}^\infty 2^{\frac{k}{3^k}}\\ \log(p) = \sum \limits_{k=1}^\infty \log\left(2^{\frac{k}{3^k}}\right) =\log(2) \sum \limits_{k=1}^\infty\dfrac{k}{3^k} = \\ \log(2) \dfrac{3}{(3-1)^2} = \log(2) \dfrac 3 4\\ e^{\log(p)} = e^{\log(2)\frac 3 4} = 2^{\frac 3 4}\)

.
 Aug 9, 2019
edited by Rom  Aug 9, 2019

8 Online Users

avatar