TITLE: Is Fractal perimeter always infinite?
QUESTION [11 upvotes]: 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 ?

REPLY [3 votes]: Yes and No.
Yes, you can generate an everywhere-bending curve like the Koch snowflake but with finite length, in exactly the way you describe.
It will be a fractal in some ways but not others, so whether it is technically a fractal will depend on who you ask.
Quoting from Wikipedia:

There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. [...] In 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way."

The curve you are looking for will not satisfy the first definition, but it can satisfy the second.  (Similarly, the answer to the title of this question is "Yes" for the first definition but "No" for the second.)
Note that these are just two of many possible definitions.  The informality of the second definition corresponds to the fact that mathematics has no real need to formally classify objects as fractals or not -- the objects themselves are the more interesting thing to study.
Properties of your curve
Your curve will necessarily have the property that as you zoom in on a random part of it, it will look more and more straight.  This is unavoidable, as otherwise the curve would indeed have infinite length.  However, it can still have "bends" everywhere, just like the Koch snowflake.  They just need to be minor bends, getting less and less sharp as you zoom in.
Related to this, your curve will have a fractal dimension of one, since you are asking for its one dimensional length to be finite.  This is why it will not satisfy the first definition above.
Another property of fractals mentioned on the Wikipedia page is that they are usually not differentiable anywhere.  Your curve, however, will be differentiable almost everywhere.  On the other hand, unlike a smooth curve, its derivative will still have discontinuities in every part of it, no matter how much you zoom in, due to its fractal nature.
An example
One easy way to make such a curve is to alter the Koch snowflake recipe so that the angles between the four segments become less sharp at each stage.  As you suggest, any convergent series will do.
So that the terms don't decrease too quickly, let's use $\sum_1^n \frac{1}{n^2}$.  Each iteration of the construction will increase the length by some multiplicative factor, so we want the logs of these factors to correspond to the terms of the convergent series, so at step $n$ we want the length to increase by a factor of $e^{1/n^2}$.
Since the terms drop quickly at the beginning, I'll skip the first 10 terms, and I'll scale the remaining terms (of the arithmetic series) by 34.81 so that the first term is roughly 4/3, like the Koch snowflake.  This gives an increase in length of $e^{\frac{34.81}{(n+10)^2}}$ at the $n$'th step, and results in an overall length of about 3.3 times the straight-line distance between the endpoints of the curve.
Here are some views of the resulting curve:
(The first one is the whole curve.)

Of course, if you would zoom in exactly on one of the "elbows" of the curve, then you would see a bend at that angle no matter how far you zoom in.