SOLUTION: For {{{ 0<x<1 }}}, the value of the expression, {{{ log(1+x)+log(1+x^2)+log(1+x^4) }}}+.... to infinity is?

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: For {{{ 0<x<1 }}}, the value of the expression, {{{ log(1+x)+log(1+x^2)+log(1+x^4) }}}+.... to infinity is?      Log On


   

Question 1032759: For +0%3Cx%3C1+, the value of the expression,
+log%281%2Bx%29%2Blog%281%2Bx%5E2%29%2Blog%281%2Bx%5E4%29++.... to infinity is?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
+...
=log(%281%2Bx%29%281%2Bx%5E2%29%281%2Bx%5E4%29%281%2Bx%5E8%29*...)
=log(1%2Bx+%2B+x%5E2+%2B+x%5E3+%2B+x%5E4+%2B+x%5E5%29+...)
=log%28%281%2F%281-x%29%29%29+=+-log%28%281-x%29%29.
The infinite geometric series 1%2Bx+%2B+x%5E2+%2B+x%5E3+%2B+x%5E4+%2B+x%5E5%29+... converges to 1%2F%281-x%29 by virtue of 0+%3C+x+%3C+1.