# Math 241 Homework 1 ```Math 241
Name: __________________________________
Homework 1
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1. One method of defining a sequence ˜+8 ™8œ&quot; is to specify the first term, and then give a recursive
formula for the subsequent terms. For example, the equations
+&quot; œ &quot;
and
+8 œ #+8&quot; for 8  &quot;
define the sequence e&quot;&szlig; #&szlig; %&szlig; )&szlig; &quot;'&szlig; &aacute; f. The first equation specifies that +&quot; is &quot;, while the second
equation says that each subsequent term +8 is equal to twice the previous term +8&quot; .
&quot;
+8&quot;  % for 8  &quot;. 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 +&quot; œ &quot; 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 ,&quot; œ ! and ,8 œ cosa,8&quot; b for 8  &quot;. Find lim ,8 , correct to
8&Auml;_
six decimal places. (Note: You do not need to show your work. Make sure to use radians!)
_
2. Consider the infinite series &quot;
8œ&quot;
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?
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