laxieqv wrote:

Is 1+x+x^2+x^3+x^4 > 1/(1-x) ?

(1) x > 0

(2) x < 1

(1) x>0 :

the LHS is always > 0

but the RHS can be <0 coz if x>1 ---> 1-x<0 ---> 1/(1-x) <0

----> we can't conclude ----> insuff

(2) x<1 :

that means x can come very very very close to 1 ----> 1-x comes very very very close to 0 ----> 1/(1-x) is very very very big. Whereas, LHS can't be too bigger than 1 since x<1 ---> all the power of x will be < x

---->we can't conclude ---> insuff

(1) and (2):

Transfer RHS all to LHS, we have the problem become:

(1+x+x^2+x^3+x^4) - 1/ (1-x) > 0

<==> [(1+x+x^2+x^3+x^4)* (1-x)]/ (1-x) - 1/(1-x) >0

we have : (1+x+x^2+x^3+x^4)* (1-x) = 1+x+x^2+x^3+x^4 - ( x+x^2+x^3+x^4+x^5) = 1-x^5

---> LHS= (1-x^5 -1) / (1-x) = -x^5 / (1-x)

since 0<x<1 ---> 0< 1-x<1 AND -x^5 < 0

----> -x^5/ (1-x) < 0

----> the answer is LHS is always < RHS

-----> suff.

C it is.