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Calculus  II                                           Quiz #10                   May 4, 2004

Name_________________            R.  Hammack                  Score ______
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Decide if the following series converge or diverge. Use any applicable test.

(1)   Underoverscript[∑ , k = 1, arg3] k^2/(k^(1/2) + k^4)
The terms are positive, and k^2/(k^(1/2) + k^4) <k^2/( k^4) = 1/( k^2),
so it converges by comparison with the convergent p-series  Underoverscript[∑ , k = 0, arg3] 1/( k^2) .

(2)  Underoverscript[∑ , k = 0, arg3] (-1)^k2/(k + 2)   
This is an alternating series that meets the conditions of the Alternating Series Test, so it CONVERGES.


(3)    Underoverscript[∑ , k = 1, arg3] (-π)^k/k !
Let's try the ratio test for absolute convergence.
ρ = Underscript[lim , k∞] (| (-π)^(k + 1)/(k + 1) ! |)/(| (-π)^k/ ... (π^(k + 1) k !)/((k + 1) ! π^k) = Underscript[lim , k∞] π/(k + 1) = 0
Since ρ<1, the original series converges absolutely, so it CONVERGES


(4)   Underoverscript[∑ , k = 1, arg3] sin(k^2 + k)/(k^2 + k)
Let's test for absolute convergence, so we look at the positive term series  Underoverscript[∑ , k = 1, arg3] | sin(k^2 + k)/(k^2 + k) |.
Now, | sin(k^2 + k)/(k^2 + k) | <1/(k^2 + k) <1/k^2, so it converges by comparison with the convergent p-series  Underoverscript[∑ , k = 0, arg3] 1/( k^2) .  
Therefore the original series is absolutely convergent, so it CONVERGES.