Tests for Sequences and Series

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Calculus

Infinite Sequences

AP Calculus BC

Unit 10: Infinite Sequences and Series

12th

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40 Terms

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Nth Term

an

<p>a<sub>n</sub></p>
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What does the nth term test for?

divergence

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Nth Term diverges conditions

lim a ≠ 0

<p>lim a <span>≠ 0</span></p>
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Geometric Series

arn

<p>ar<sup>n</sup></p>
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Geometric Series Convergence Conditions

l r l < 1

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Geometric Series Divergence Conditions

l r l ≥ 1

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What does the Geometric Series converge to?

the sum; S = a / (1-r)

<p>the sum; S = a / (1-r)</p>
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Telescoping Series

(an - an+1)

<p>(a<sub>n</sub> - a<sub>n+1</sub>)</p>
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Telescoping Series Convergence Conditions

lim an = L; L has to be a finite number

<p>lim a<sub>n</sub> = L; L has to be a finite number</p>
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What does the telescoping Series converge to?

the sum; S = a1 - L

<p>the sum; S = a<sub>1 </sub>- L</p>
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P-series test

1 / np

<p>1 / n<sup>p</sup></p>
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P-series test convergence conditions

p > 1

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P-series test divergence conditions

0 < p ≤ 1

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What if p = 1 in a p-series test?

divergent harmonic series

<p>divergent harmonic series</p>
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Alternating Series Test

(-1)n * an or (-1)n+1 * an+1

<p>(-1)<sup>n</sup> * a<sub>n</sub> or (-1)<sup>n+1</sup> * a<sub>n+1</sub></p>
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Alternating Series Test convergent conditions

0 < an+1 ≤ an (nonincreasing) & lim an = 0

<p>0 &lt; a<sub>n+1</sub> ≤ a<sub>n</sub> (nonincreasing) &amp; lim a<sub>n</sub> = 0</p>
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Alternating Series Test Remainder

l S - Sn l = l Rn l ≤ an+1

<p>l S - S<sub>n</sub> l = l R<sub>n</sub> l <span>≤ a<sub>n+1</sub></span></p>
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Integral Test conditions for f(x)

positive, continuous, and decreasing

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Integral Test

an = f(n)

<p>a<sub>n</sub> = f(n)</p>
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Integral Test Convergence Conditions

∫ f(x) dx converges, then series converges

<p><span>∫ f(x) dx converges, then series converges</span></p>
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Integral Test Divergence Conditions

∫ f(x) dx diverges, then series diverges

<p>∫ f(x) dx diverges, then series diverges</p>
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Ratio Test

an

<p>a<sub>n</sub></p>
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Ratio Test Convergence Conditions

lim l an+1 / an l < 1

<p>lim l a<sub>n+1</sub> / a<sub>n</sub> l &lt; 1</p>
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Ratio Test Divergence Conditions

lim l an+1 / an l > 1 or = ∞

<p>lim l a<sub>n+1</sub> / a<sub>n</sub> l &gt; 1 or = <span>∞</span></p>
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Ratio Test is inconclusive if?

lim l an+1 / an l = 1

<p>lim l a<sub>n+1</sub> / a<sub>n</sub> l = 1</p>
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Root Test

an

<p>a<sub>n</sub></p>
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Root Test Convergence Conditions

lim n√l an l < 1

<p>lim n√l a<sub>n</sub> l &lt; 1</p>
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Root Test Divergence Conditions

lim n√l an l > 1

<p>lim n√l a<sub>n</sub> l &gt; 1</p>
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The root test is inconclusive when?

lim n√l an l = 1

<p>lim n√l a<sub>n</sub> l = 1</p>
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Direct Comparison Test

an

<p>a<sub>n</sub></p>
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Direct Comparison Test Convergence Conditions

0 < an ≤ bn and bn converges

<p>0 &lt; a<sub>n</sub> <span>≤ b<sub>n</sub> and b<sub>n</sub> converges</span></p>
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Direct Comparison Test Divergence Conditions

0 < bn ≤ an and bn diverges

<p>0 &lt; b<sub>n</sub> ≤ a<sub>n</sub> and b<sub>n</sub> diverges</p>
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Limit Comparison Test

an

<p>a<sub>n</sub></p>
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Limit Comparison Test Convergence Conditions

lim (an / bn) = L > 0 & bn converges

<p>lim (a<sub>n </sub>/ b<sub>n</sub>) = L &gt; 0 &amp; b<sub>n </sub>converges</p>
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Limit Comparison Test Divergence Conditions

lim (an / bn) = L > 0 & bn diverges

<p>lim (a<sub>n </sub>/ b<sub>n</sub>) = L &gt; 0 &amp; b<sub>n </sub>diverges</p>
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L must be what in limit comparison test?

finite and positive

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Absolute vs. Conditional Convergence

once identified as convergence, use tests to see if l an l is convergent; yes- absolute or no- conditional

<p>once identified as convergence, use tests to see if l a<sub>n</sub> l is convergent; yes- absolute or no- conditional</p>
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Sequence

a list; ex: 2, 4, 6, 8

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Series

ex: 2 + 4 + 6 + 8

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Recursive Definition (Implicit Definition) vs. Explicit Definition for Sequences

recursive- an+1 = an + x; explicit- an = 3n - 2