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  1. 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

  2. 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.

  3. 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 + … .

  4. Partial Sums An infinite series contains many terms. Sequence of partial sums:

  5. 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.

  6. 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

  7. (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

  8. Arithmetic Series Summing the terms of a arithmetic sequence results in an arithmetic series.

  9. 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,

  10. Arithmetic Series Since an = a1 + (n – 1)d, Sn can also be written in the following way.

  11. 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

  12. 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

  13. 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

  14. 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

  15. Geometric Series The sum of the terms of a geometric sequence is called a geometric series.

  16. 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

  17. Approximate the sum for the given values of n. Solution Example: Finding the sum of finite geometric series

  18. Example: Finding the sum of finite geometric series

  19. 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

  20. 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

  21. 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.

  22. 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

  23. 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.

  24. Evaluate each series. Solution Example: Using summation notation

  25. Example: Using summation notation

  26. Properties for Summation Notation Let a1, a2, a3, …, an and b1, b2, b3, …, bn be sequences, and c be a constant.

  27. Properties for Summation Notation Let a1, a2, a3, …, an and b1, b2, b3, …, bn be sequences, and c be a constant.

  28. Use properties for summation notation to find each sum. Solution Example: Applying summation notation

  29. Example: Applying summation notation