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
  • Alternating in sign, and
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:  

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