# Geometric series/Ratio test/R/Section

The series is called *geometric series* for
,
so this is the sum

The convergence depends heavily on the modulus of .

The following statement is called *ratio test*.

Let

be a series of real numbers. Suppose there exists a real number with , and a with

for all (in particular for ). Then the series converges absolutely.

The convergence does not change (though the sum) when we change finitely many members of the series. Therefore, we can assume . Moreover, we can assume that all are positive real numbers. Then

Hence, the convergence follows from the comparison test and the convergence of the geometric series.

The *Koch snowflakes* are given by the sequence of plane geometric shapes , which are defined recursively in the following way: The starting object is an equilateral triangle. The object is obtained from by replacing in each edge of the third in the middle by the corresponding equilateral triangle showing outside.

Let denote the area and the length of the boundary of the -th Koch snowflake. We want to show that the sequence converges and that the sequence diverges to .

The number of edges of is , since in each division step, one edge is replaced by four edges. Their length is of the length of a previous edge. Let denote the base length of the starting equilateral triangle. Then consists of edges of length and the length of all edges of together is

Because of , this diverges to .

When we turn from to , there will be for every edge a new triangle whose side length is a third of the edge length. The area of an equilateral triangle with side length is . So in the step from to there are triangles added with area . The total area of is therefore

If we forget the and the factor , which does not change the convergence property, we get in the bracket a partial sum of the geometric series for , and this converges.