I apologise beforehand for the possible errors in my post, I am an undergraduate student in Greece and I am translating Greek terminology in to english sort of on the go. Also excuse me if this questions sounds trivial but I am a beginner in this subject.

I am a physics student and taking a numerical analysis (calculus?) course and our professor gave us this problem after having recently done the Picard method else known as $x=g(x)$ method as he taught it to us (I am not asking anyone to do my homework for me, I just have a question of mathematical nature and I don't know if it's my own misunderstanding or an actual problem):

We have the function $f(x)=e^{2x}-3x-1$ and we need to find its roots with said method, how we do this and the starting points we choose are left to our own discretion. 

Looking for the root $x=0$ specifically I solved for $x$ as such: 

$$x=\frac 13e^{2x}-\frac 13=g(x)$$ 

and started iterations and according to the professor we should stop iterations when the value 

$$\epsilon\_n=\left|\frac{x\_{n+1}-x\_n}{x\_{n+1}}\right| < 0.01$$

for this specific problem. 

Although the method should converge this way (this can be proven) I see that the value $\epsilon\_n$ actually increases with each iteration and slowly converges at the value $\epsilon\_n=0.5$.

My question being: is this correct? i.e. Is this supposed to happen, or is it a misunderstanding/miscalculation of my own in this problem?

Again very sorry if this sounds trivial or like I'm just assigning my homework to smarter people since this isn't really what I'm trying to do here.