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14.2.6.1: Prelude to Sequence and Series

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    The Koch snowflake1 is constructed from an infinite number of non-overlapping equilateral triangles. Consequently, we can express its area as a sum of infinitely many terms. How do we add an infinite number of terms? Can a sum of an infinite number of terms be finite? To answer these questions, we need to introduce the concept of an infinite series, a sum with infinitely many terms. Having defined the necessary tools, we will be able to calculate the area of the Koch snowflake.

    This is a diagram of several iterations of the Koch snowflake, which is created through an interative process. The first case is an equilateral triangle. Five times, the middle third of each line segment is replaced with an equilateral triangle pointing outward.
    Figure \(\PageIndex{1}\): The Koch snowflake is constructed by using an iterative process. Starting with an equilateral triangle, at each step of the process the middle third of each line segment is removed and replaced with an equilateral triangle pointing outward.

    The topic of infinite series may seem unrelated to differential and integral calculus. In fact, an infinite series whose terms involve powers of a variable is a powerful tool that we can use to express functions as “infinite polynomials.” We can use infinite series to evaluate complicated functions, approximate definite integrals, and create new functions. In addition, infinite series are used to solve differential equations that model physical behavior, from tiny electronic circuits to Earth-orbiting satellites.

    1The Koch snowflake is a Koch curve which is a fractal curve. Fractals are seen in nature (snowflake for instance as well as trees, coastlines, broccoli, galaxies, etc.) but are also seen in constructions such as the Khajuraho temples in India (see below). Fractal analysis (many methods which are usually used together for robust comparison) which is an analysis that looks for fractal characteristics to extract information form a signal. One application of fractal analysis is cancer detection.

    Note that there are different types of fractals. The Koch snowflake is fractal curve that lends itself to the discussion in this section, other fractals are not applicable to this section. Fractals are real mathematics/science but there are a numerous pseudo-science writings that use fractals in questionable  ways. Follow the guidelines discussed herein before when reading about fractals.

    Khajuraho_Group_of_Monuments_4.jpg
    A couple of the Khajuraho temples which have a fractal structure to them. Wikipedia image (Abinthomas0007, CC BY-SA 4.0,9/30/2017).

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