Term
| Geometric Series Test (GST) |
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Definition
If the series is geometric with constant ratio r, then
the series converges when [image],
and the series diverges when [image] |
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Definition
If [image] is a series of positive terms, then:
the series converges when [image],
the series diverges when [image], and
the test is inconclusive when [image]. |
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Term
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Definition
[image] diverges if [image].
The test is inconclusive when [image]. |
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Term
| Absolute Convergence Test (ACT) |
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Definition
If [image] converges, then [image] converges.
If [image] diverges, then the test is inconclusive. |
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Term
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Definition
If [image] is a series of positive terms, and [image], where p is a constant, then:
the series converges when [image] and
the series diverges when [image] |
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Term
| Direct Comparison Test (DCT) |
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Definition
Let [image] be a known series, and [image] and [image] be series of positive terms. Then:
[image] converges if [image] for all [image] and [image] converges.
[image] diverges if [image] and [image] diverges.
Otherwise the test is inconclusive. |
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Term
| Limit Comparison Test (LCT) |
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Definition
Let [image] and [image] be series of positive terms.
If [image], where c is non-zero, then both series either converge or diverge.
If [image] and [image] converges, then [image] conerges as well.
If [image], and [image] diverges, then [image] diverges as well. |
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Term
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Definition
Let [image] be a series of positive terms, where [image], such that [image] is positive, decreasing, and continuous for all [image] for some positive integer N, then
[image] and [image] either both converge or both diverge. |
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Term
| Alternating Series Test (AST) |
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Definition
Let [image] be:
1) an alternating series
2) whose terms decrease in absolute value
3) such that [image].
Then [image] converges.
If any of the above three coniditions is false, the test is inconclusive. |
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Term
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Definition
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Term
| Use the Ratio Test when... |
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Definition
| the series is non-geometric and nothing else works, or if there are factorials involved. |
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Term
| Use the nth-Term Test when... |
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Definition
| the terms of the series are obviously not going to zero. |
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Term
| Use the Absolute Convergence Test when... |
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Definition
| taking the absolute value of a series would create a convenient series of positive terms. This test is almost never used on its own. Rather it is often combined with another test (like Ratio Test) |
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Term
| Use the P-series Test when... |
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Definition
| the series is a p-series. Combine with ACT as needed. |
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Term
| Use the Direct Comparison Test when... |
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Definition
| the series is either obviously larger than a simple divergent series, or obviously smaller than a simpler convergent series. |
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Term
| Use the Limit Comparison Test when... |
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Definition
| the EBM of the series is easily understood. |
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Term
| Use the Integral Test when... |
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Definition
| you can solve the corresponding improper integral. |
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Term
| Use the Alternating Series Test when... |
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Definition
| the series is alternating and its terms are obviously getting closer to zero. |
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