Math 241 Homework 1

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 "
+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
six decimal places. (Note: You do not need to show your work. Make sure to use radians!)
2. Consider the infinite series "
(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?