a+ar+ar^2 + ...The term of the series is ar^n.This series converges if |r|<1 and diverges otherwise.If it converges, it converges to a/(1-r). A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence i.e a sequence of numbers in which the ratio between consecutive terms is constant. 16. Partial sums of geometric series Start (how else?) with partial sums: A nite geometric sum is of the form: S N = a + ar + ar2 + ar3 + + arN Multiply both sides by r to get: rS N = ar + ar2 + ar3 + ar4 + + arN+1 Now subtract the second equation from the rst (look at all the cancellation on the right side!) For this series nd, (b) the common ratio,  (c) the rst term, [2â¦ A new series, obtained by squaring each term of the original series, has sum 16 times the sum of the original series. I'm not too sure how to go about answering this question. To determine the long-term effect of Warfarin, we considered a finite geometric series of $$n$$ terms, and then considered what happened as $$n$$ was allowed to grow without bound. For example, the series For any two successive terms in the geometric series Î£ar^(n-1), the ratio of the two terms, (ar^n) / ar^(n-1), simplifies into an algebraic expression given by? A geometric series is any series that we can write in the form $a+ar+ar^2+ar^3+â¯=\sum_{n=1}^âar^{nâ1}.$ Because the ratio of each term in this series to the previous term is r, the number r is called the ratio. $$\{a, ar, ar^2, ar^3, ar^4, \ldots\}$$ The sum of all the terms, is called the summation of the sequence. Before we can learn how to determine the convergence or divergence of a geometric series, we have to define a geometric series. Geometric Series. Once you determine that youâre working with a geometric series, you can use the geometric series test to determine the convergence or divergence of the series. Definition 8.2.2. geometric sequence: An ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. A geometric series is a+ ar + ar2 + ::: (a) Prove that the sum of the rst n terms of this series is given by S n = a(1 rn) 1 r  The third and fth terms of a geometric series are 5.4 and 1.944 respectively and all the terms in the series are positive. 9 - 11 + 121/9 ... is a geometric series. Each term of a geometric series, therefore, involves a higher power than the previous term. Series List Geometric Series. In this sense, we were actually interested in an infinite geometric series (the result of letting $$n$$ go to infinity in the finite sum). An infinite geometric series has sum 2000. The general form of a geometric sequence is: $a, ar, ar^2, ar^3, ar^4, \cdots$ ... Key Terms. The common ratio of the original series is m/n, where m and n are relatively prime positive integers. We refer to a as the initial term because it is the first term in the series. Also known as a geometric â¦ The summation of an infinite sequence of values is called a series . Geometric Series A pure geometric series or geometric progression is one where the ratio, r, between successive terms is a constant.