![]() ![]() (BOTTOM) Gaps filled by broadening and decreasing the heights of the separated trapezoids.Īfter knowing that a series converges, there are some applications in which it is also important to know how quickly the series converges. (MIDDLE) Gaps caused by addition of adjacent areas. (TOP) Alternating positive and negative areas. Rate of convergence Converging alternating geometric series with common ratio r = -1/2 and coefficient a = 1. Let’s look at some examples of sequences. In the limit, as the number of trapezoids approaches infinity, the white triangle remainder vanishes as it is filled by trapezoids and therefore s n converges to s, provided | r|1, the trapezoid areas representing the terms of the series instead get progressively wider and taller and farther from the origin, not converging to the origin and not converging as a series. In this course we will be interested in sequences of a more mathematical nature mostly we will be interested in sequences of numbers, but occasionally we will nd it interesting to consider sequences of points in a plane or in space, or even sequences of sets. ![]() The trapezoid areas (i.e., the values of the terms) get progressively thinner and shorter and closer to the origin. Each additional term in the partial series reduces the area of that white triangle remainder by the area of the trapezoid representing the added term. The area of the white triangle is the series remainder = s − s n = ar n+1 / (1 − r). For example, the series 1 2 + 1 4 + 1 8 + 1 16 + ⋯ Īlternatively, a geometric interpretation of the convergence is shown in the adjacent diagram. In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. 1 In mathematics, the inequality of arithmetic and geometric means, or more briefly the AMGM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same. The total purple area is S = a / (1 - r) = (4/9) / (1 - (1/9)) = 1/2, which can be confirmed by observing that the unit square is partitioned into an infinite number of L-shaped areas each with four purple squares and four yellow squares, which is half purple. Taking square roots and dividing by two gives the AMGM inequality. Another geometric series (coefficient a = 4/9 and common ratio r = 1/9) shown as areas of purple squares. The sum of the areas of the purple squares is one third of the area of the large square. It alsoexplores particular types of sequence known as arithmetic progressions (APs) and geometricprogressions (GPs), and the corresponding series. Each of the purple squares has 1/4 of the area of the next larger square (1/2× 1/2 = 1/4, 1/4×1/4 = 1/16, etc.). Arithmetic and geometric progressions Arithmetic andgeometric progressions mcTY-apgp-2009-1 This unit introduces sequences and series, and gives some simple examples of each. ![]() Sum of an (infinite) geometric progression The geometric series 1/4 + 1/16 + 1/64 + 1/256 +. ![]()
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