Activity Title: Exploring Sequences, Series, Convergence and Divergence Using EXCEL

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Activity Title: Exploring Sequences, Series, Convergence and Divergence using EXCEL

Materials List: MS Excel, Sequences and Series Spreadsheet Template (included in the zip file)

Procedure:

Background: Using formulas in an Excel Spreadsheet is an effective way to model Arithmetic and Geometric Sequences and Series. Each Sequence can be graphed to illustrate the relationships to linear and exponential functions. In Excel we can also examine the principal of convergence of an Infinite Geometric Series. Students will be able to see the series or sum is approaching a known limit as it goes towards infinity.

Preparation: Have the Sequences and Series Spreadsheet Template ready for students

Arithmetic Formulas:

Sequence:

Series:

Geometric Formulas:

Sequence:

Series:

Infinite Series:

Using Excel Formulas: To complete this module you will be asked to develop excel formulas that will create the desired sequence or series. All Excel formulas begin with an “=” sign. If you want to use a cell as part of your formula, you can click on that cell to insert it. For example, “=2*C14” will multiply 2 and cell C14. Some helpful operations:

 +, -, *, /  () can be used for order of operations  “^” to raise to an exponent  “sqrt” for square root

of 4 Lab Activity:

1. Open the Sequences and Series Spreadsheet Template provided by the instructor. 2. Notice that the spreadsheet has 5 sheet tabs that you will progress through. 3. Make sure you are on the first tab “Arithmetic P1.” On this sheet a sample arithmetic sequence and series are provided. a) Click on cell B3 and record the formula: ______b) This is getting a value from another cell, what is this value for? ______c) Notice the dollar sign in the middle of the cell name, this keeps the formula fixed on this cell for all the cells in the column. d) Click on cell C4, what is this formula doing? ______e) Now write a formula that finds the nth term of any arithmetic sequence if the first term and common difference cells in F6 and F8 are changed along with a value inserted cell H10 for N. (Hint: remember that arithmetic sequences are linear.) Test your formula to make sure it works for different values. f) Record your formula: ______4. Click on tab 2 “Arithmetic P2.” On this sheet you are given two terms and their corresponding values. (Hint: think of ordered pairs)

a) In cell F9 write a formula to find the common difference for this sequence. (Hint: think slope) Record your formula: ______

b) In cell F11 write a formula to find the first term. Make sure it would work if the values in the cells change. Record your formula: ______

c) In cell F13 write a formula to find the nth term, given a value for n in cell H13. Make sure it works if the given values for term and value are changed. Record your formula: ______

5. Click on tab 3 “Geometric P1.” On this sheet a sample geometric sequence and series are provided.

a) Click on cell B4, what is this doing? ______b) Now write a formula that finds the nth term of any geometric sequence if the first term and common ratio cells in F6 and F8 are changed along with a value inserted cell H10 for N. (Hint: remember that geometric sequences are exponential.) Test your formula to make sure it works for different values. c) Record your formula: ______

6. Click on tab 4 “Geometric P2.” On this sheet you are given two terms and their corresponding values.

a) In cell G7 write a formula to find the common ratio for this sequence. (Use hints given on the sheet, you may want to Google how to find the common ratio from 2 values.) Record your formula: ______

b) In cell G9 write a formula to find the first term. Make sure it would work if the values in the cells change. Record your formula: ______

c) In cell F11 write a formula to find the nth term, given a value for n in cell H11. Make sure it works if the given values for term and value are changed. Record your formula: ______

7. Click on tab 5 “Divergence and Convergence”. This sheet gives you a sample geometric sequence and series.

a) Note that the common ratio is 3. Observe the series column, at 20 terms we are at a sum of over 10 billion. As this series continues it will approach: ______

In math we would say that this series diverges or is divergent.

b) Change the common ratio in cell F6 from 3 to 1/3. Observe the series column again. As it approaches 20 terms, the sum is approaching the number: ______

In math we would say that this series converges or is convergent because it will reach a finite solution.

c) Change the first term in cell F8 to 10. Then change the common ratio to the values given in the table below. Observe the value that the series is approaching, this is the sum of each infinite geometric series. Record the sums in the table below.

of 4 Initial Term Common Ratio Sum of Infinite Series 10 1/2 10 1/3 10 1/4 10 1/5 10 2/3 10 3/4 10 4/5 d) Look for a pattern. Can you explain how to find the sum of any infinite geometric series given the initial term and common ratio. If needed try other initial terms and ratios.

______