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or 2n 5 3n 2 A 2n 7 3n 5 , 6n2 11n 10 A 6n2 11n 35, i.e. 10A 35, which is true. Thus, by reversal of steps in the inequalities, we see that fun g is monotonic increasing. Actually, since 10 > 35, the sequence is strictly increasing. (b) By writing some terms of the sequence, we may guess that an upper bound is 2 (for example). To prove this we must show that un @ 2. If 2n 7 = 3n 2 @ 2 then 2n 7 @ 6n 4 or 4n < 11, which is true. Reversal of steps proves that 2 is an upper bound. (c) Since this particular sequence is monotonic increasing, the rst term 1 is a lower bound, i.e., un A 1, n 1; 2; 3; . . . . Any number less than 1 is also a lower bound. Thus, for example, we can write

First select 2 * 3: 2 *3 + 5 Then press [CTRL] + [SHIFT] + [ENTER] (PC) or [COMMAND] + [RETURN] (Mac). 6+5 A semicolon (;) at the end of a Mathematica command will suppress output. This is useful in long sequences of calculations when only the final answer is important.

EXAMPLE 3 Suppose we wish to define a = 1, b = 2, c = 3 and then display their sum. Here are two ways to write

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(d) Since the sequence has an upper and lower bound, it is bounded. jun j @ 2 for all n. (e)

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Occasionally you may introduce an instruction that takes an excessively long time to execute, or you may inadvertently create an infinite loop. To abort a calculation, go to Evaluation Abort Evaluation. Alternatively, you may press [ALT] + [ . ] to abort ([COMMAND] + [ . ] on the Macintosh). On the rare occasion when this does not work, you will have to terminate the kernel by going to Evaluation Quit Kernel Local. However, by doing so, you will lose all your defined symbols and values. Your Mathematica notebook will not be lost, however, so they can easily be restored. As with all computer software, there are times when Mathematica will crash completely. The only remedy is to close Mathematica and reload it. On rare occasions, you may have to reboot your computer. In either event, your notebook changes will be lost. It is therefore extremely important to back up your notebook often! Finally, there may be times when you wish to include comments within your Mathematica commands. Anything written within (* and *)is ignored by the Mathematica kernel.

12 + (* these words will be ignored by the kernel *) 3 15

1.3 The following program is an infinite loop. Execute it and then abort the evaluation. x = 1; While[x > 0, x = x + 1]

Since every bounded monotonic (increasing or decreasing) sequence has a limit, the given sequence has 2n 7 2 7=n 2 a limit. In fact, lim lim . n!1 3n 2 n!1 3 2=n 3

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17.2 * (16.3 + 4.7) 361.2

Mathematica is case sensitive. For example, Integrate and integrate are different. All Mathematica-defined symbols, commands and functions begin with a capital letter. Some symbols, such as FindRoot, use more than one capital letter. To avoid conflicts, it is a good idea for all user-defined symbols to begin with a lowercase letter. Different brackets are used for different purposes.

Square brackets are used for function arguments: Sin[x] not Sin(x).

2.16. A sequence fun g is de ned by the recursion formula un 1 exists. (b) Find the limit in (a).

Lists are general objects that contain collections of other objects. In reading this chapter you will see that lists are used for a variety of applications. Therefore, Mathematica offers an extensive collection of list manipulation commands. The objects within a list are contained within curly brackets, {}. Alternatively, the List command may be used to define a list.

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