Is Fractal perimeter always infinite? Looking for information on fractals through google I have read several time that one characteristic of fractals is : 


*

*finite area

*infinite perimeter


Although I can feel the area is finite (at least on the picture of fractal I used to see, but maybe it is not necessarly true ?), I am wondering if the perimeter of a fractal is always infinite ?
If you think about series with positive terms, one can find :


*

*divergent series : harmonic series for example $\sum_0^\infty{\frac{1}{n}}$

*convergent series :  $\sum_0^\infty{\frac{1}{2^n}}$


So why couldn't we imagine a fractal built the same way we build the Koch Snowflake but ensuring that at each iteration the new perimeter has grown less than $\frac{1}{2^n}$ or any term that make the whole series convergent ?
What in the definition of fractals allows or prevent to have an infinite perimeter ?

If they have infinite sides, than they must have an infinite perimeter, especially if they are perfectly straight because the formula of perimeter of most shapes is adding up the amount of sides, and the fractal has infinite sides, then it should have an infinite perimeter.  
