Understanding Series in Mathematics: Definition, History, and Applications, Study notes of Mathematics

Download Understanding Series in Mathematics: Definition, History, and Applications and more Study notes Mathematics in PDF only on Docsity! What is series? A series an is the indicated sum of all values of an when n is set to each integer from a to b inclusive; namely, the indicated sum of the values aa + AA+1 + AA+2 + ... + ab-1 + ab. Definition of the "Sum of the Series": The "sum of the series" is the actual result when all the terms of the series are summed. In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. A series in math is simply the sum of the various numbers, or elements of a sequence. For example, to make a series from the sequence of the first five positive integers 1, 2, 3, 4, 5, just add them up. So, 1 + 2 + 3 + 4 + 5 = 15 is a series.   Let the terms in a series be denoted by the symbol, an , and the nth partial summation be denoted using the following sigma notation for any natural number n: Example If we have a sequence 1, 4, 7, 10, … Then the series of this sequence is 1 + 4 + 7 + 10 +… The Greek symbol sigma “Σ” is used for the series, which means “sum up”.  The series 4 + 8 + 12 + 16 + 20 + 24 can be expressed as Applications of series in real life ∑i=164n∑i=164n We read this expression as the sum of 4n as n ranges from 1 to 6.  For any sequence, its series can be calculated by summing up its numbers. History of the theory of infinite series Development of infinite series Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.[15][16] Mathematicians from Kerala, India studied infinite series around 1350 CE.[17] In the 17th century, James Gregory worked in the new decimal system on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series. Convergence criteria The investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence. Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form. Difference between Sequences and Series Sequences Series Set of elements that follow a pattern Sum of elements of the sequence Order of elements is important Order of elements is not so important Finite sequence: 1,2,3,4,5 Finite series: 1+2+3+4+5 Infinite sequence: 1,2,3,4,…… Infinite Series: 1+2+3+4+…… Applications of series in real life Sequences and series are very important in mathematics and also have many useful applications, in areas such as finance, physics and statistics. It's always better to know how knowledge helps us in real life. Stacking cups, chairs, bowls etc. ... Pyramid-like patterns, where objects are increasing or decreasing in a constant manner. ... Filling something is another good example. ... Seating around tables. ... Fencing and perimeter examples are always nice. 1. Musical harmonics When you pluck a string on a musical instrument, it creates more than one note. It creates a base note and also a collection of higher notes called harmonics or the (musical) harmonic series.The different musical harmonics correspond to the different terms of the mathematical harmonic series. When a string is plucked, it vibrates along its whole length to form the base note. At the same time, it vibrates in two pieces to form the first harmonic of the musical harmonic series. It vibrates in three pieces to form the next harmonic of the musical harmonic series. And so on, and so on. If we say the original length of the string is 1, then the base note and harmonics of the musical harmonic series correspond to the lengths These are the terms of the mathematical harmonic series 2. Waiting time for bus An arithmetic sequence can help you calculate the time you will have to wait before the next bus arrives 3. Business analysis Mathematical sequences and series are also used in business and financial analysis to assist in decision-making and find the best solution to a given problem. ... If you are a business analyst, a statistician or an investment manager, your work will revolve around number patterns and the analysis of these patterns. 4. In Making Pyramid-like Structures