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      <text>8.4.2 (History of sequences and series)</text>
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      <text><![CDATA[<h3>History of sequences and series</h3>
<p>As with many of the topics we have encountered so far in the course, the origins of sequences and series can be traced back to antiquity. However, it is their role as the beginning of mathematical analysis which best demonstrates their importance to scientific study. Mathematical analysis is a relatively young topic, whose beginnings lie in Europe in the 16th and 17th centuries. Analysis is a branch of mathematics that studies situations with continuous change, such as the study of motion and the geometry of smooth curves and surfaces. It transpires that these ideas require the computation of infinite series and sequences as we have seen this week.</p>
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<p>Perhaps the earliest systematic study of series came with the ancient Greeks, via the paradoxes of Zeno. Little is known about Zeno and we only know of his works via Aristotle who often quoted them simply to refute them. Zeno listed a number of paradoxes about motion which puzzled mathematicians for centuries, but were eventually solve by summing infinite series. Just as Pythagoras' discovery of the number \(\sqrt{2}\) challenged the Greeks' ideas of what a number actually is, Zeno's paradoxes challenged their idea of what motion is. Zeno was not the only Greek mathematician to work on series and sequences, one of the first major advancements in the area came with the discovery of the method of exhaustion. This was developed by Eudoxus and was a method to calculate the area or volume of any shape by defining them as combinations of multiple polyhedra. For example, the idea was that we could measure the area of a circle by calculating the areas of polygons with ever increasing number of sides (See figure below). This example was first shown by the Chinese mathematician Liu Hui (circa 300CE). The Greek knowledge of the method of exhaustion was extended by Archimedes who applied it, to find volumes and areas of a wide variety of shapes and curves.</p>
<p><br /> <img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/method%20of%20exhaustion.png" alt="area of a circle by exhaustion" width="621" height="207" /></p>
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<p>The next major developments in the study of sequences and series are intertwined with the development of differential calculus which we shall examine in week 10. However we should make a small detour to mention the work of Fibonacci (1180-1250CE) whose actual name was Leonardo Pisano Bigollo, Fibonacci meaning 'son of Bonacci'. Fibonacci was one of the first European thinker to use he Arabic system of numerals \(0\) to \(9\) which we use today. The famous sequence which bears his name was invented in response to a problem about rabbit breeding, which we will introduce later in this section. Actually, Fibonacci was responsible for several advances in number theory, but his contributions are often somewhat overlooked, and how he is remembered mostly for his famous sequence.</p>
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<p>Some of the first infinite series were studied in the 14th century by Indian mathematician Madhava of Sangamagrama, who gave an infinite series form of the trigonometric functions. He inspired a school of followers in Kerala who dedicated themselves to expanding his works up to the 16th century. In the 16th and 17th centuries, Europe saw the beginning of differential calculus, and with it the foundations of analysis, which began to be used as a formal language for the calculus. Many fmous mathematicians contributed to its development, from Descartes and Fermat who discovered analytic geometry to the prolific Euler, who introduced the idea of a mathematical function and led to Bolzano's (in 1816CE) definition of continuity. This in turn influenced the work of Cauchy who introduced the idea of complex analysis. This was then expanded by Liouville and Fourier who studied harmoinc analysis, which is a topic based around series of the form \(\sum\limits_{n=1}^\infty \frac{1}{n}\) which is called the <em>harmonic series</em> and governs the study of harmonics in both mathematics and music. In the 19th century Riemann gave a definition of integration, based upon adding infinite areas. This eventually led to the formalisation of calculus using set theory in the 20th century, this finally led to the subject of Functional analysis, created by Hilbert and Banach. This remains a very active area of research in the mathematical community to this day.</p>]]></text>
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