prove that 9 power 1/3 + 9 power 1/9 + 9 power 1/27 +............ infinity = 3?? how to do this????

Your question is wrong.

It should be like :

9^1/3 * 9^1/9 * 9^1/27 *... *infinite = 3.

So it would change into

9^{1/3 + 1/9 + 1/27 + ...}

Now  sequence in the power is ,

1/3 + 1/9 + 1/27 + .....upto infinity

It is a infinite GP with

a = 1/3 , r = 1/3 

So sum of infinite term of GP is

= a/(1-r)

= (1/3 ) / (1 - 1/3)

=(1/3) / (2/3)

= 1/2

so we have

=9^1/2

=3