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 ?
You can certainly define a fractal analogous to the Koch snowflake with finite perimeter.

The iterative rule for the standard Koch snowflake is to replace the middle third of each line segment with two line segments of equal length and meeting the original segment and each other at $60^\circ$ angles.  This applies the rule $x \mapsto 4x/3$ to the overall length of the curve, so applying it repeatedly produces a divergent sequence of lengths.

A modified rule might be to start with a red line segment and iteratively replace the middle half of each red line segment with the top of a trapezoid made of one red and two blue line segments (all of equal length and again making $60^\circ$ angles).  Then the overall red/blue lengths $(x,y)$ are subjected to the rule $(x,y)\mapsto(3x/4, y + x/2)$, which is convergent when iterated.  It is a nice exercise to show that this process, starting from a red segment of length $1$, produces an entirely blue curve of length $2$; and starting from an equilateral red triangle of perimeter $3$ produces a closed blue curve of perimeter $6$.  (Calculating the area inside the curve is another nice exercise.  Moreover, since the trapezoid can be constructed either red-blue-blue or blue-red-blue at each step, there are any number of different fractals with the same perimeter and area.)
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.