BdMO Online Forum

Solve the following equation :
\[\frac{2}{3}\left [ 1+\frac{1}{1+x} +\frac{1}{(1+x)^3}+\frac{1}{(1+x)^5 }+\cdot \cdot \cdot\right ]=1\]

I can't write latex well. so I am giving short solution with banglish.( can't know the english of some word)
bracket ar vitorer ongshe osim gunnottor dharar sutro proyog korle :
3(X^2+2X) = 2(1+2X+X^2)
and ata solve korle
X = -(1+root3) or root3 -1 answer ase
If i haven't done any mistake

notice that the sequence inside the bracket is not a geometric sequence unless you cut out $1$ from it.
now do it in the same way. if my calculation is not wrong
\[x=\pm\sqrt2\]

*edited.courtesy:protik

my mistake. my answer is also \[\pm\] \[x=\sqrt2\]

Solved

here.