#3^(1/3)× 9^(1/9) × 27^(1/27) xx cdots xx 3^(k/(3^k)) xx cdots = #?

1 Answer
Nov 24, 2017

See below.

Explanation:

#Pi=3^(1/3)× 9^(1/9) × 27^(1/27) xx cdots xx 3^(k/(3^k)) xx cdots#

#Pi = 3^(1/3+2/9+3/27 + cdots k/3^k + cdots) = 3^I#

now for #absx < 1#

#sum_(k=0)^oo x^k = 1/(1-x)# and

#sum_(k=0)^oo k x^k = x d/(dx)(1/(1-x)) = x/(1-x)^2#

now making #x = 1/3 rArr I = (1/3)/(1-1/3)^2 = 3/4# and finally

#Pi = 3^(3/4)#