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## Series

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**Series**11.2 Understand basic concepts about series Identify and find the sum of arithmetic series Identify and fid the sum of geometric series Learn to sue summation notation**Introduction**A series is the summation of the terms in a sequence. Series are used to approximate functions that are too complicated to have a simple formula. Series are instrumental in calculating approximations of numbers like π and e.**Series**A finite series is an expression of the form a1 + a2 + a3 + … + an , and an infinite series is an expression of the form a1 + a2 + a3 + … + an + … .**Partial Sums**An infinite series contains many terms. Sequence of partial sums:**Partial Sums**If Sn approaches a real number S as ng∞ then the sum of the infinite series is S. For example, let S1 = 0.3, S2 = 0.3 + 0.03, S3 = 0.3 + 0.03 + 0.003 and so on. Some infinite series do not have a sum S. For example, the series given by1 + 2 + 3 + 4 + 5 + …would have an unbounded, or “infinite,” sum.**For each an, calculate S4.**(a)an = 2n + 1 (b) an = n2 Solution Since S4 = a1 + a2 + a3 + a4, start calculating the first four terms of the sequence an = 2n + 1. a1 = 2(1) + 1 = 3 a3 = 2(3) + 1 = 7 a2 = 2(2) + 1 = 5 a4 = 2(4) + 1 = 11 Thus S4 = 3 + 5 + 7 + 9 = 24 Example: Finding partial sums**(b) an = n2**a1 = 12 = 1 a3 = 32= 9 a2 = 22 = 4 a4 = 42 = 16 Thus S4 = 1 + 4 + 9 + 16 = 30 Example: Finding partial sums**Arithmetic Series**Summing the terms of a arithmetic sequence results in an arithmetic series.**Sum of the First n Terms of an Arithmetic Sequence**The sum of the first n terms of an arithmetic sequence, denoted Sn,is found by averaging the first and nth terms and then multiplying by n. That is,**Arithmetic Series**Since an = a1 + (n – 1)d, Sn can also be written in the following way.**Use a formula to find the sum of the arithmetic series**2 + 4 + 6 + 8 + …+ 100. Solution The series has n = 50 terms with a1 = 2 and a50 = 100. We can use Example: Finding the sum of a finite arithmetic series**A person has a starting annual salary of $30,000 and**receives a $1500 raise each year. (a)Calculate the total amount earned over 10 years. (b)Verify this value using a calculator. Example: Finding the sum of a finite arithmetic series**Solution**(a) The arithmetic sequence describing the salary during year n is computed by an = 30,000 + 1500(n – 1). The first and tenth year’s salaries are a1 = 30,000 + 1500(1 – 1) = 30,000 a10 = 30,000 + 1500(10 – 1) = 43,500 Example: Finding the sum of a finite arithmetic series**Solution**(b) To verify this result with a calculator, compute the sum a1+a2+a3+…+a10, where an= 30,000 + 1500(n – 1). Theresult of 367,500 agrees with part (a). Example: Finding the sum of a finite arithmetic series**Geometric Series**The sum of the terms of a geometric sequence is called a geometric series.**Sum of the First n Terms of an Geometric Sequence**If a geometric sequence has first term a1 and common ratio r, then the sum of the firstn terms is given by**Approximate the sum for the given values of n.**Solution Example: Finding the sum of finite geometric series**Annuities**An annuity is a sequence of deposits made at equal periods of time. After n years the amount is given by This is a geometric series with first terma1 = A0 and common ratio r = (1 + i). The sum of the first n terms is given by**Suppose that a 20-year-old worker deposits $1000 into an**account at the end of each year until age 65. If the interest rate is 4%, find the future value of the annuity. SolutionLet A0 = 1000, i = 0.04, andn = 45. The future value of the annuity is given by Example: Finding future value of an annuity**Sum of an Infinite Geometric Series**The sum of the infinite geometric sequence with first term a1 and common ratio r isgiven by Provided |r| < 1. If |r| ≥ 1, then this sum does not exist.**Find the sum of the infinite geometric series**Solution a1 = 1, common ratio is r = 0.5 Example: Finding the sum of an infinite geometric series**Summation Notation**Summation notation is used to write series efficiently. The symbol , sigma, indicates the sum. The letter k is called the index of summation. The numbers 1 and n represent the subscripts of the first and last term in the series.They are called the lower limit and upper limit of the summation, respectively.**Evaluate each series.**Solution Example: Using summation notation**Properties for Summation Notation**Let a1, a2, a3, …, an and b1, b2, b3, …, bn be sequences, and c be a constant.**Properties for Summation Notation**Let a1, a2, a3, …, an and b1, b2, b3, …, bn be sequences, and c be a constant.**Use properties for summation notation to find each sum.**Solution Example: Applying summation notation