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Understanding Sequences and Series in Mathematics

Explore sequence, factorial, and summation notation as tools for writing and evaluating terms and sums in mathematics. Learn how to find patterns, express nth terms, and work with finite and infinite sequences and series effectively.

andrewdmartin
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Understanding Sequences and Series in Mathematics

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  1. 8.1 Sequences and Series Essential Questions: How do we use sequence notation to write the terms of a sequence? How do we use factorial notation? How do we use summation notation to write sums?

  2. A sequence is a list of numbers written in an explicit order. nth term Any real-valued function with domain a subset of the positive integers is a sequence. If the domain has end-points, then the sequence is a finite sequence. In calculus, we will mostly be concerned with infinite sequences.

  3. Definition of a Sequence An infinite sequence is a function whose domain is the set of positive integers. The function values are the terms of the sequence. If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

  4. The first four terms of the sequence given by are Finding Terms of a Sequence • The first four terms of the sequence given by are

  5. Finding Terms of a Sequence • Write out the first five terms of the sequence given by Solution:

  6. Apparent pattern: Each term is 1 less than twice n, which implies that Apparent pattern: Each term is 1 more than the square of n, which implies that Finding the nth term of a Sequence • Write an expression for the apparent nth term (an) of each sequence. • a. 1, 3, 5, 7, … b. 2, 5, 10, 17, … Solution: • n: 1 2 3 4 . . . n terms: 1 3 5 7 . . . an • n: 1 2 3 4 … n terms: 2 5 10 17 … an

  7. Apparent pattern: Each term has a numerator that is 1 greater than its denominator, which implies that Additional Example • Write an expression for the apparent nth term of the sequence: Solution:

  8. Factorial Notation • If n is a positive integer, n factorial is defined by As a special case, zero factorial is defined as 0! = 1. Here are some values of n! for the first several nonnegative integers. Notice that 0! is 1 by definition. The value of n does not have to be very large before the value of n! becomes huge. For instance, 10! = 3,628,800.

  9. Finding the Terms of a Sequence Involving Factorials • List the first five terms of the sequence given by Begin with n = 0.

  10. Evaluating Factorial Expressions • Evaluate each factorial expression. Make sure you use parentheses when necessary. a. b. c. Solution: a. b. c.

  11. Have you ever seen this sequence before? • 1, 1, 2, 3, 5, 8 … • Can you find the next three terms in the sequence? • Hint: 13, • 21, 34 • Can you explain this pattern?

  12. The Fibonacci Sequence • Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. A well-known example is the Fibonacci Sequence. • The Fibonacci Sequence is defined as follows: Write the first six terms of the Fibonacci Sequence:

  13. Using Σ notation For example: When working with series, the Greek symbol Σ (the capital letter sigma) is used to mean ‘the sum of’. … and this is the last value of r. This is the first value of r … represents a finite series containing n terms: u1 + u2 + u3 + … + un The terms in the series are obtained by substituting 1, 2, 3, …, n in turn for r in ur.

  14. Summation Notation • Definition of Summation Notation The sum of the first n terms of a sequence is represented by Where i is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation.

  15. To Find a Sum with Sigma • Plug in each consecutive value from the lower limit to the upper limit into the equation. • Add each term together.

  16. Summation Notation for Sums • Find each sum. a. b. c. Solution: a.

  17. Notice that this summation is very close to the irrational number . It can be shown that as more terms of the sequence whose nth term is 1/n! are added, the sum becomes closer and closer to e. Solutions continued b. c.

  18. How to Input Sums in your calculator Be sure you are in sequence mode on the calculator! • Good news! This can all be done using the TI-84 Plus graphing calculator. • To enter in example a, hit the following keys: The following screen will appear: Now hit 5. Then, hit • Good news! This can all be done using the TI-84 Plus graphing calculator. • To enter in example a, hit the following keys: The following screen will appear: Now hit 5. Then, hit The following screen should appear: Choose 5.

  19. This is what your calculator screen should look like: Now type in the sum, the variable, the lower limit, the upper limit, and the increment (default is 1).

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