# Inequality test returned unevaluated

I tried,

Refine[Sin[k]^2/(k^2 + k + 1) < 1/(k^2), Element[k, Integers]]

$\frac{\text{Sin}[k]^2}{1+k+k^2}<\frac{1}{k^2}$

It returns unevaluated. I tried converting Sin[x] into Taylor series but it still doesn't evaluate it.Is there some work around for this or is it that I am doing something wrong ?

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Mathematica returns this inequality:

In[1]:= ineq = Sin[k]^2/(k^2 + k + 1) < 1/(k^2);

unevaluated for integer k because it can be true, false or meaningless for different integers.

For positive (real not only integer number) it's true:

In[2]:= Refine[ineq, k > 0]
Out[2]= True

For 0 RHS doesn't exist and e.g. for -2 it's false:

In[3]:= ineq /. k -> -2
Out[3]= False
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answer is correct, I got my mistake, I didn't mention domain for K. –  Rorschach Jun 13 '14 at 5:13