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Unformatted text preview: 9 mcs-ftl 2010/9/8 0:40 page 243 #249 Sums and Asymptotics Sums and products arise regularly in the analysis of algorithms, financial applica- tions, physical problems, and probabilistic systems. For example, we have already encountered the sum 1 C 2 C 4 C C N when counting the number of nodes in a complete binary tree with N inputs. Although such a sum can be represented compactly using the sigma notation log N X 2 i ; (9.1) i D it is a lot easier and more helpful to express the sum by its closed form value 2N 1: By closed form , we mean an expression that does not make use of summation or product symbols or otherwise need those handy (but sometimes troublesome) dots. . . . Expressions in closed form are usually easier to evaluate (it doesnt get much simpler than 2N 1 , for example) and it is usually easier to get a feel for their magnitude than expressions involving large sums and products. But how do you find a closed form for a sum or product? Well, its part math and part art. And it is the subject of this chapter. We will start the chapter with a motivating example involving annuities. Figuring out the value of the annuity will involve a large and nasty-looking sum. We will then describe several methods for finding closed forms for all sorts of sums, including the annuity sums. In some cases, a closed form for a sum may not exist and so we will provide a general method for finding good upper and lower bounds on the sum (which are closed form, of course). The methods we develop for sums will also work for products since you can convert any product into a sum by taking a logarithm of the product. As an example, we will use this approach to find a good closed-form approximation to n WWD 1 2 3 n: We conclude the chapter with a discussion of asymptotic notation. Asymptotic notation is often used to bound the error terms when there is no exact closed form expression for a sum or product. It also provides a convenient way to express the growth rate or order of magnitude of a sum or product. 244 mcs-ftl 2010/9/8 0:40 page 244 #250 Chapter 9 Sums and Asymptotics 9.1 The Value of an Annuity Would you prefer a million dollars today or $50,000 a year for the rest of your life? On the one hand, instant gratification is nice. On the other hand, the total dollars received at $50K per year is much larger if you live long enough. Formally, this is a question about the value of an annuity. An annuity is a finan- cial instrument that pays out a fixed amount of money at the beginning of every year for some specified number of years. In particular, an n-year, m-payment annuity pays m dollars at the start of each year for n years. In some cases, n is finite, but not always. Examples include lottery payouts, student loans, and home mortgages....
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