![]() Want to cite, share, or modify this book? This book uses the We can substitute a 1 = 50, r = 1.005, and n = 72 a 1 = 50, r = 1.005, and n = 72 into the formula, and simplify to find the value of the annuity after 6 years. In 6 years, there are 72 months, so n = 72. We can find the value of the annuity after n n deposits using the formula for the sum of the first n n terms of a geometric series. ![]() Let us see if we can determine the amount in the college fund and the interest earned. After the first deposit, the value of the annuity will be $50. We can find the value of the annuity right after the last deposit by using a geometric series with a 1 = 50 a 1 = 50 and r = 100.5 % = 1.005. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. The account paid 6% annual interest, compounded monthly. This is the value of the initial deposit. In the example, the parent invests $50 each month. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. ![]() An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. ∑ k = 1 ∞ ( − 3 8 ) k ∑ k = 1 ∞ ( − 3 8 ) k Solving Annuity ProblemsĪt the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month into a college fund for six years. If the terms of an infinite geometric sequence approach 0, the sum of an infinite geometric series can be defined. Determining Whether the Sum of an Infinite Geometric Series is Defined When the sum is not a real number, we say the series diverges. Therefore, the sum of this infinite series is not defined. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. This series can also be written in summation notation as ∑ k = 1 ∞ 2 k, ∑ k = 1 ∞ 2 k, where the upper limit of summation is infinity. An example of an infinite series is 2 + 4 + 6 + 8 +. An infinite series is the sum of the terms of an infinite sequence. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first n n terms. Thus far, we have looked only at finite series. How much will she have earned by the end of 8 years? Using the Formula for the Sum of an Infinite Geometric Series We can begin by substituting the terms for k k and listing out the terms of this series.Īt a new job, an employee’s starting salary is $32,100. If we interpret the given notation, we see that it asks us to find the sum of the terms in the series a k = 2 k a k = 2 k for k = 1 k = 1 through k = 5. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. A variable called the index of summation is written below the sigma. An explicit formula for each term of the series is given to the right of the sigma. Summation notation includes an explicit formula and specifies the first and last terms in the series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, Σ, Σ, to represent the sum. Summation notation is used to represent series. Consider, for example, the following series. The sum of the terms of a sequence is called a series. To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. To do so, we need to consider the amount of money invested and the amount of interest earned. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. The fund pays 6% annual interest, compounded monthly. They plan to invest $50 in the fund each month.
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