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| What is the rule for Convergence and Divergence of a P-Series? |
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Definition
[image]
Convergent if P > 1
Divergent if P [image] 1 |
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| When is a Geometric Series Convergent or Divergent? |
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Definition
[image]
Convergent |r| < 1
Divergent |r| [image] 1 |
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| When can you use the Comparison Test? |
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Definition
if an ≤ bn for all n, and ∑bn is convergent then ∑an is Convergent
if an ≥ bn for all n, and ∑bn divergent, then ∑an is divergent |
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| What is the Divergence test? How do you use it? |
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Definition
| lim an ≠ 0 Then the ∑an is divergent |
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| What is the Integral test? |
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Definition
if ∫f(x)dx converges, then the series converges
If it diverges then the series diverges |
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Definition
lim|(an+1)/(an)|
if the lim >1 it is divergent
if the lim < 1 it is convergent
if the lim = 1 then the test is inconclusive |
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Definition
If for all n, an is positive, non-increasing (i.e. 0 < an+1 <= an), and approaching zero, then the alternating series
sum (1..inf) (-1)n an and sum (1..inf) (-1)n-1 an
both converge.
If the alternating series converges, then the remainder RN = S - SN (where S is the exact sum of the infinite series and SN is the sum of the first N terms of the series) is bounded by |RN| <= aN+1 |
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Definition
Let L = lim (n -- > inf) | an |1/n.
If L < 1, then the series sum (1..inf) an converges.
If L > 1, then the series sum (1..inf) an diverges.
If L = 1, then the test in inconclusive. |
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