geometric_series

# geometric_series - Geometric Series Before we define what...

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G eometric S eries Before we define what is meant by a series, we need to introduce a related topic, that of sequences. Formally, a sequence is a function that computes an ordered list. Suppose that on day 1, you have 1 dollar, and every day you double your money. Then the function f ( n ) = 2 n generates the sequence 1, 2, 4, 8, 16, 32, …, when n = 1, 2, 3, 4, 5, 6, … This list represents the amount of dollars you have after n days. Note: The use of “…” is read as “and so on”. The individual entries in a sequence are called the terms of the sequence. In our discussion, we are going to assume that the terms in a particular sequence are real numbers. Sequences can be grouped into two large classes based upon the number of terms they include. An infinite sequence is a function that has the set of natural numbers as its domain. As the name implies, it contains an infinite number of terms. In the opening example, the use of the “…” without some number on the end implies that the sequence continues indefinitely, following the prescribed pattern. Of course, there is an inherent problem with assuming that money can be doubled forever. Instead, it makes sense to talk about doubling money for a certain number of days. Say, for n = 1, 2, 3, 4, 5, 6, and 7. In that case, the sequence generated would be called a finite sequence . Its domain is equal to a finite set of natural numbers. (In this case, D = {1, 2, …, 7}.) A common notation for sequences is let a n = f ( n ). With this notation, we say that a n is the n th term in the sequence. Example 1: Write out the first five terms a 1 , a 2 , a 3 , a 4 and a 5 of the following sequences. (a) (1 ) n n a  (b)  2 sin n an (c) 25 n (d) 1 2(3) n n a 1

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Solution: (a) a 1 = (-1) 1 = -1, a 2 = (-1) 2 = 1, a 3 = (-1) 3 = -1, a 4 = (-1) 4 = 1, a 5 = (-1) 5 = -1. (b)  1 22 sin (1) sin 1 a   ,   2 2 sin (2) sin 0 a , 3 3 sin (3) sin 1 a ,   4 2 sin (4) sin 2 0 a , 5 5 sin (5) sin 1 a . (c) , , 1 2(1) 5 3 a  2 2(2) 5 1 a 3 2(3) 5 1 a  , , 4 2(4) 5 3 a 5 2(5) 5 5 a . (d) , 11 0 1 2(3) 2(3) 2 a 21 1 2 2(3) 2(3) 6 a , 31 2 3 2(3) 2(3) 18 a , 41 3 4 2(3) 2(3) 54 a , . 51 4 5 2(3) 2(3) 162 a It is worth noting that using these formulas we would easily compute the 1,000 th term in the sequence. We would only need to plug in n = 1000.
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geometric_series - Geometric Series Before we define what...

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