009C Sample Midterm 3, Problem 4
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Test the series for convergence or divergence.
 (a) (6 points)
 (b) (6 points)
Foundations: 

For , both sine and cosine of are strictly nonnegative. Thus, these series are alternating, and we can apply the 
Alternating Series Test: If a series is 


then the series is convergent. 
Note that if the series does not converge to zero, we must claim it diverges by the 
Divergence Test: If then the series/sum diverges. 
In the case of an alternating series, such as the two listed for this problem, we can choose to show it does not converge to zero absolutely. 
Solution:
(a): 

Here, we have 

(b): 

Final Answer: 
