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 then may or may not converge.
Hint: Ask yourself: How does go to zero? In the limit, does resemble a familiar sequence? Does the familiar series have known convergence properties? If so, you have the beginnings of a strategy for showing convergence or divergence.
PSeries
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Geometric Series and related tests.
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 This is the granddaddy of many series which are easy to sum. It also is the foundation for several other tests when you observe:
 The ratio of successive terms of the Geometric Series is x  hence the Ratio Test
 The ratio of the absolute values of successive terms of the Geometric Series is x  hence the Ratio Test for Absolute Convergence
 The n^{th} root of the n^{th} term of the Geometric Series is x  hence the Root Test
Ratio Test
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be a series with positive terms and then:
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 Try this test when u_{k} involves factorials or k^{th} powers.
Ratio Test for Absolute Convergence
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be a series with nonzero terms and then:
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 The series need not have positive terms and need not be alternating to use this test since any series converges if it converges absolutely.
Root Test
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be a series with positive terms and then:
Comment: Try this test when u_{k} involves k^{th} powers.
Integral Test
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be a series with positive terms and let f(x) be the function that results when k is replaced by x in the formula for u_{k}. If f is a decreasing, continuous function for x > N then: have like convergence (either both converge or both diverge).
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 Use this test when f(x) is easy to integrate.
Limit Comparison Test
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 be series with positive terms such that:
then the two series have like convergence (either both converge or both diverge). then notice which series "won".
 Your unknown series converges if it is clearly smaller than a convergent series  think about it.
 Your unknown series diverges if it is clearly larger than a divergent series  think about it.
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 This is easier to apply than the Comparison Test, but still requires some skill in choosing the known series. The Divergence Test can be a great source of inspiration here.
