Introduction to Sequences in Maple (Classic Version for Windows)

Transcription

1 Introduction to Sequences in Maple (Classic Version for Windows) Author: Barbara Forrest Contact: Copyrighted/NOT FOR RESALE version 1.1 Contents 1 Objectives for this Lab ii 2 Helpful Hints for using Maple ii 3 Manipulating Expressions in Maple iii 4 Generating the Terms of a Sequence in Maple iv 5 Plotting the Terms of a Sequence in Maple v 6 Finding the Limit of a Sequence in Maple vi 7 Practice Exercises vii i

2 1 Objectives for this Lab Review the contents of A Short Introduction to Maple. Read about helpful hints for using Maple. Learn how to expand, factor, and solve expressions in Maple. Listing the terms of a sequence in Maple. Plotting the terms of a sequence in Maple. Finding the limit of a sequence in Maple. 2 Helpful Hints for using Maple What to do when things go wrong? You have seen 1 that the [> restart: command tells Maple to clear its memory. It is always a good idea to begin a worksheet with this command. If you ever find that you are entering commands within a worksheet, but Maple is either returning what seems to be incorrect answers or is doing nothing, the best action to take is to enter the restart command again. Also, if you find that Maple is taking too long to do a calculation or that you want Maple to stop performing a calculation, click the Stop icon found in the toolbar. In some versions of Maple, the Stop icon may appear as. If these methods do not work, save your work (if possible), quit Maple and restart the Maple program to continue working. To avoid any loss of work, you should save your worksheet often. Start Maple CLASSIC version. Begin a new worksheet by entering the restart: command. Avoiding too much typing? One of the drawbacks of using the CLASSIC version of Maple is that commands must be typed at the command prompt. 2 To avoid too much typing, remember to use copy and paste. Enter the following command at a Maple prompt: [> 2 8*(sqrt(3) + sin(pi/4)); Once you have entered this expression, you realize that instead you wanted to use 5. You could retype the command at the next Maple prompt. However, a faster method is to copy and paste the expression, and then edit. 1. Use your mouse to select the expression above, 2 8*(sqrt(3) + sin(pi/4)); (i.e., click and drag to highight the expression). 2. From the Edit menu, choose Copy (or alternatively, press CTRL + C on your keyboard). 1 Review the document entitled A Short Introduction to Maple if you haven t already. 2 Recall that the Standard version of Maple uses an enhanced worksheet with many user-friendly tools. However, the Classic version uses less computer memory. See?versions for more information. ii

3 3. Click your mouse cursor beside the next available Maple prompt. 4. From the Edit menu, choose Paste (or alternatively, press CTRL + V on your keyboard). 5. Move the cursor to 3 and change this number to 5. Press ENTER. You should copy and paste expressions whenever possible to save time. Including comments with Maple commands. In the Short Introduction to Maple you have already seen how to include a text region in a Maple worksheet. You can also include comments beside Maple commands to document your work. The number sign (#) is used to insert a comment. Enter the following at a Maple prompt and then press ENTER: [> 1/4 + 1/2; # This is my first comment. We are adding fractions. Notice that anything after the # sign is ignored by Maple. You should comment your work by using text regions and by using comments so your worksheet is easier to follow. The most common Maple CLASSIC syntax error. Maple uses the following symbols +,,, \, and for addition, substraction, multiplication, division, and exponentiation, respectively. It is a common error to forget to include the in an expression when multiplying. For example, [> 2x + 3; #This is incorrect since the * is missing. Maple returns an error. [> 2*x + 3; #This is correct. Enter both of the previous expressions to see how Maple CLASSIC reacts to a syntax error. 3 Manipulating Expressions in Maple Three tasks that are often required in algebra are expanding, factoring, and solving an expression. Expanding an expression. Suppose you wanted to expand the expression (2x 2 6x + 11)(3x 2 + 7x 12). You can easily find the answer in Maple by using the expand command. Try entering the following in Maple: [> (2*x 2-6*x + 11)*(3*x 2 + 7*x - 12); # Maple returns what you have typed. Now try using the expand command. Type the following (remember to use copy and paste to save some typing!) [> expand((2*x 2-6*x + 11)*(3*x 2 + 7*x - 12)); # Maple expands the expression by multiplying. This is much easier than doing the computation by hand! Factoring an expression. You can factor an expression using Maple s factor command. If the expression cannot be factored in Maple, the original expression is returned. Try entering the following in Maple: [> factor(8*x 3-12*x 2 + 6*x - 1); # Maple factors the expression. [> factor(8*x 3-12*x 2 + 3*x - 1); # Maple can t factor the expression. Solving an expression. Solving an expression means to find all the values of the unknown (usually the unknown is x) so that the expression is equal to 0. You can solve an expression using Maple s solve command. If the expression cannot be solved in Maple, nothing is returned. Try entering the following in Maple: iii

4 [> solve(2*x *x 2+19*x-9.6); # Maple solves the expression. Maple informs us that the values of x that will make the expression 2x x x 9.6 equal to 0 are x = 1, x = 1.5, and x = 3.2. Note that Maple uses 10 significant digits in the answer. If Maple can t solve the expression, nothing is returned. Try entering the following. [> solve(1/(x-1)); # Maple can t solve the expression. This expression can never be equal to 0 (i.e., the numerator can never be 0), so Maple returns nothing. You can even solve expressions that are inequalities in Maple. For example, solve x 2 5 < 4. Enter the following: [> solve(x 2-5 < 4); Maple returns ( 3, 3) This means the solution is the open interval from 3 to 3. In the next example, note that abs represents the absolute value of an expression and <= means less than or equal to. This time let s try to solve x 5 2. [> solve(abs(x-5) <= 2); Maple returns [3, 7] Alternately, the solution to [> solve(abs(x-5) < 2); is (3, 7) 4 Generating the Terms of a Sequence in Maple To list the terms of a sequence {a n } in Maple you must use the seq command. The general syntax for the seq command is: seq( sequence formula, n = index bounds ) For example, let a n = n2 3 n. To generate the first 10 terms of this sequence, try entering the following command in a Maple 2 worksheet: [ > seq( n 2 / (3 - n 2), n = ); Maple returns the following output: 1/2, 4, 3/2, 16/13, 25/22, 12/11, 49/46, 64/61, 27/26, 100/97 You can modify this command to include the index number of each term by using the syntax seq( [n, sequence formula ], n = index bounds ) iv

5 For example, again let a n = n2 3 n. To generate the first 10 terms of this sequence along with the index of each term, try 2 entering the following command: [ > seq( [n, n 2 / (3 - n 2)], n = ); Maple returns the following output: [1, 1/2], [2, 4], [3, 3/2], [4, 16/13], [5, 25/22], [6, 12/11], [7, 49/46], [8, 64/61], [9, 27/26], [10, 100/97] List the first 20 terms of the sequence a n = 1 n. List the first 20 terms of the sequence a n = 1 n again, but this time list each term with its index number. 5 Plotting the Terms of a Sequence in Maple Recall that you saw how to plot basic functions in the Short Introduction to Maple. The syntax to plot the terms of a sequence is similar. To plot the terms of the sequence against the index of the terms, we will use a new variable named data and the Maple plot command. Note that you do not have to use data to name the set of terms of the sequence; you can use any other name with alphanumeric characters. The general syntax to assign the label data to the sequence terms in Maple is: data := seq( [ n, sequence formula ], n = index range of data points ) Then you would enter the following command to plot each term of the sequence on the Real plane: [ > plot( [data], style = point); For example, try entering the following commands in a Maple worksheet to plot the first 10 terms: [ > data := seq( [ n, n 2 / (3 - n 2) ], n = ); [ > plot( [data], style = point ); Note that you can modify this plot command so that the terms appear as different symbols. For example, try: v

6 [ > plot( [data], style = point, symbol = circle ); The symbol option in Maple s plot command will accept asterisk, box, circle, cross, diagonalcross, diamond, point, solidbox, solidcircle, or soliddiamond. You can try another symbol type in your plot command to see how the plotted terms will appear. For example, try [ > plot( [data], style = point, symbol = solidbox ); If you are interested, see the?plot,options help page in Maple for more information about plot options in Maple. Plot the first 20 terms of the sequence a n = 1 n. Edit the previous exercise and plot the first terms of the sequence a n = 1 n. Edit the previous exercise and plot only the terms from n = 9500 to n = Hint: Use plot([seq([n,1/n],n= )],style=point); 6 Finding the Limit of a Sequence in Maple To find the limit of a sequence you can use Maple s limit command and the following syntax: limit( sequence formula, n = index bound ); For example, let a n = n2 3 n. To find lim a n, try entering the following command: 2 n [ > limit( n 2 / (3 - n 2), n = infinity ); Maple returns the limit as 1. You should check that this is the correct answer by doing the hand calculation. Calculate lim cos(n) in Maple. Consider the result. What is Maple trying to tell you? Try plotting the first 100 x terms of the sequence a n = cos(n). Does Maple s limit result match the plot? vi

7 7 Practice Exercises Complete the following exercises in a Maple worksheet and write down your answers in the space provided below. Save the worksheet as sequence.mws. Questions from these practice exercises may appear on the course assignments. 1. Create the following text input. At the top of a new Maple worksheet, type the following lines in a text region. Maple Lab : Sequences Name: ID: Date: Type your name here. Type your ID number here. Type today s date here. You can enhance the fonts in any way you wish (e.g., change the font, fontsize, bold, etc.) Insert an execution group after the last line of the text region. Enter Maple s restart: command. 2. Compute the following in Maple. Write your answers in the space beside each question. Leave your answers in symbolic form. (a) (b) (Be careful with your brackets!) (c) ( ) 1 2 (d) Compute the floating-point (decimal) approximation of the following expressions in Maple. (Hint: use the evalf command.) Write your answers in the space beside each question. (a) (b) (c) Solve the following expressions in Maple. Leave your answers in symbolic form. Write your answers in the space beside each question. (a) 2x 2 + 3x 100 (b) (x 6) (3x+4) (2x 3) (x+2) (c) 4 3x 2 5. Let a n = cos(n)2 2 n. Find the limit of this sequence in Maple. Create the plot of the first 20 terms. Did your calculated limit match the Maple plot of the sequence terms? 6. Let a n = n! 2 n. Create the plot of the first 1000 terms. Can you guess the limit of the sequence? Find the limit of this sequence in Maple. 7. (a) Let a n = cos(n 2 ). Generate the first 10 terms in Maple. Can you guess the limit of the sequence by inspecting these terms? (b) Let a n = cos(n 2 ). Create the plot of the first 1000 terms. Can you guess the limit of the sequence? Find the limit of this sequence in Maple. What is the Maple output trying to tell you? 8. Save the worksheet. Save the worksheet as sequence.mws. You may print the worksheet if you would like a copy for your notes. To print a copy of the worksheet, from the File menu, choose Print and click OK. vii