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3 May 2019 Before we can learn how to determine the convergence or divergence of a geometric series, we have to define a geometric series. Once you

(a) A geometric series has first term 𝑎 and common ratio 𝑟. Prove the sum of the first 𝑛 terms is given by S n= 𝑎(1−𝑟𝑛) 1−𝑟. (3) (b) The fourth term of a geometric series is -108 and seventh term is 4. (i) Find the common ratio of the series. Geometric Series - Proof of the Sum of the first n terms : ExamSolutions. Watch later.

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Geometric series, in mathematics, an infinite series of the form a + ar + ar2 + ar3 +⋯, where r is known as the common ratio. A simple example is the geometric series for a = 1 and r = 1/2, or 1 + 1/2 + 1/4 + 1/8 +⋯, which converges to a sum of 2 (or 1 if the first term is excluded). So this is a geometric series with common ratio r = –2. (I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2.) The first term of the sequence is a = –6. Plugging into the summation formula, I get: Infinite Geometric Series To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, S = a1 1 − r, where a1 is the first term and r is the common ratio. Example 6: A geometric series is the sum of the terms in a geometric sequence.

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## A geometric series is a series or summation that sums the terms of a geometric sequence. There are methods and formulas we can use to find the value of a geometric series. It can be helpful for understanding geometric series to understand arithmetic series, and both concepts will be used in upper-level Calculus topics.

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### Use this step-by-step Geometric Series Calculator, to compute the sum of an infinite geometric series providing the initial term a and the constant ratio r.

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The more general case of the ratio a rational function of the summation index produces a series called a hypergeometric series. For the simplest case of the ratio equal to a constant, the terms are of the form. Geometric series, in mathematics, an infinite series of the form a + ar + ar2 + ar3 +⋯, where r is known as the common ratio. A simple example is the geometric series for a = 1 and r = 1/2, or 1 + 1/2 + 1/4 + 1/8 +⋯, which converges to a sum of 2 (or 1 if the first term is excluded).

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### A geometric series is a series or summation that sums the terms of a geometric sequence. There are methods and formulas we can use to find the value of a

1. Recommended resources for finding nth partial sums of series.

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4. -2-6 - 18 - 54, n = 9. =1. S-1-16-4)*.