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Math 241 Name: __________________________________ Homework 1 _ 1. One method of defining a sequence ˜+8 ™8œ" is to specify the first term, and then give a recursive formula for the subsequent terms. For example, the equations +" œ " and +8 œ #+8" for 8 " define the sequence e"ß #ß %ß )ß "'ß á f. The first equation specifies that +" is ", while the second equation says that each subsequent term +8 is equal to twice the previous term +8" . " +8" % for 8 ". Make a table showing $ the first ten terms of this sequence, correct to four decimal places. (a) Let e+8 f be the sequence defined by +" œ " and +8 œ (b) Based on your data, does the sequence in part (a) converge or diverge? Explain. (c) Let e,8 f be the sequence defined by ," œ ! and ,8 œ cosa,8" b for 8 ". Find lim ,8 , correct to 8Ä_ six decimal places. (Note: You do not need to show your work. Make sure to use radians!) _ 2. Consider the infinite series " 8œ" 8 . $8 (a) Make a table showing decimal values for the first ten partial sums of this series, including at least four decimal places for each value. (You should feel free to use a spreadsheet or computer program, in which case you must attach a printout of your table.) (b) Based on your data table, what would you guess is the sum of the series?