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Divergence Test Integral
Test Comparison
Test Ratio
Test Root
Test
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test
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STATEMENT |
COMMENTS |
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Divergence Test |
If lim uk ¹0, then Suk diverges.
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If lim uk
= 0, Suk
may or may not converge. |
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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STATEMENT |
COMMENTS |
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Integral Test |
Let Suk 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 uk. If f is decreasing and continuous for x ³ 1, then ¥
+ ¥
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Use this test when f(x) is easy to integrate. This test only applies to series that have positive terms. |
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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STATEMENT |
COMMENTS |
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Comparison Test |
Let åak and åbk be series with nonnegative terms such that a1 £ b1, a2 £ b2 ,…, ak £ bk ,…. If åbk converges, then åak converges, and if åak diverges, then åbk diverges.
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Use this test as a last resort; other tests are often easier to apply. This test only applies to series with nonnegative terms. |
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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STATEMENT |
COMMENTS |
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Ratio Test |
Let åuk be a series with positive terms and suppose
uk +1 (a) Series converges if r < 1. (b) Series diverges if r > 1 or r = + ¥. (c) No conclusion if r = 1. |
Try this test when uk involves factorials or kth powers. |
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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STATEMENT |
COMMENTS |
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Root Test |
Let åuk
be a series with positive terms such that
r
= lim
(a) Series converges if r < 1. (b) Series diverges if r > 1 or r = + ¥. (c) No conclusion if r = 1. |
Try this test when uk involves kth powers. |
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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STATEMENT |
COMMENTS |
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Limit Comparison Test |
Let åak and åbk be series with positive terms such that
ak If 0 < r < + ¥, then both series converge or both diverge.
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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STATEMENT |
COMMENTS |
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Alternating Series Test |
The series (a) a1 > a2 > a3 …. (b)
lim ak = 0 |
This test applies only to alternating series. It is assumed
that ak > 0 for all k. |
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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STATEMENT |
COMMENTS |
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Ratio Test for Absolute
Convergence |
Let åuk
be a series with
nonzero terms such that:
(a)
Series converges absolutely if
r
<1 |
The series need not have positive terms and need not be alternating to use this test. |
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Note
about absolute convergence: |
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If |
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converges, then so does | ||
That is, every absolutely convergent series converges. |
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Divergence Test
Integral
Test Comparison
Test Ratio
Test Root
Test Limit Comparison Test Alternating Series Test Ratio Test for Absolute Convergence
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