Calc BC Convergence Tests

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15 Terms

1
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What is the p-series test?

If the series is of the form 1/(n^p),

If p<1, than diverges

If p>1, converges

<p>If the series is of the form 1/(n^p),</p><p>If p&lt;1, than diverges</p><p>If p&gt;1, converges</p>
2
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What is the ratio test?

Use the ratio test if have n!

do L=lim l an+1 / anl

n→∞

If lim < 1, absolutely converges

If lim > 1, diverges

If lim = 1, test fails

<p>Use the ratio test if have n!</p><p>do L=lim l an+1 / anl</p><p>n→∞</p><p>If lim &lt; 1, absolutely converges</p><p>If lim &gt; 1, diverges</p><p>If lim = 1, test fails</p>
3
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What is the root test?

Take the lim n->inf of the nth root of a[n]

If lim < 1, absolutely converges

If lim > 1, diverges

If lim = 1, test fails

<p>Take the lim n-&gt;inf of the nth root of a[n]</p><p>If lim &lt; 1, absolutely converges</p><p>If lim &gt; 1, diverges</p><p>If lim = 1, test fails</p>
4
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What is the Alternating series test?

Converges if and only if

1) lim an = 0

n→∞

2) an+1≤an

<p>Converges if and only if</p><p>1) lim an = 0</p><p>n→∞</p><p>2) an+1≤an</p>
5
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What is the Geometric Series Test?

Converges to S=a/(1-r) if lrl<1

Diverges if lrl≥1

If the series is of the form r^n

Plug in 1st # to get first term.

<p>Converges to S=a/(1-r) if lrl&lt;1</p><p>Diverges if lrl≥1</p><p>If the series is of the form r^n</p><p>Plug in 1st # to get first term.</p>
6
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What must the series do in order to use the integral test?

1. Continuous

2. Positive

3. Decreasing

7
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What is the integral test?

If the definite integral from some a to infinity of the series converges, than the series converges.

If the integral diverges, than the series diverges.

*just switch the sigma with an integral

NOTE: Very often this involves u sub, actually almost always Remember LIATE

<p>If the definite integral from some a to infinity of the series converges, than the series converges.</p><p>If the integral diverges, than the series diverges.</p><p>*just switch the sigma with an integral</p><p>NOTE: Very often this involves u sub, actually almost always Remember LIATE</p>
8
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What is the direct comparison test to converge?

If a[n] > b[n], and if a[n] converges, than b[n] must also converge.

<p>If a[n] &gt; b[n], and if a[n] converges, than b[n] must also converge.</p>
9
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What is the direct comparison test to diverge?

If a[n] < b[n], and if a[n] diverges, than b[n] must also diverge.

<p>If a[n] &lt; b[n], and if a[n] diverges, than b[n] must also diverge.</p>
10
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What is the limit comparison test?

Take the lim n->inf of a[n]/b[n]

If the lim is positive and finite, then both a[n] and b[n] either converge or diverge.

<p>Take the lim n-&gt;inf of a[n]/b[n]</p><p>If the lim is positive and finite, then both a[n] and b[n] either converge or diverge.</p>
11
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What is the absolute convergence test?

If Ia[n]I converges, than a[n] absolutely converges

<p>If Ia[n]I converges, than a[n] absolutely converges</p>
12
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nth term test (divergence test)

If the lim n -> inf of a[n] = anything but 0, it for sure diverges.

But if it does = 0, than the test fails.

hint: if the top power is teh same or greater than the bottome power than series diverges!

<p>If the lim n -&gt; inf of a[n] = anything but 0, it for sure diverges.</p><p>But if it does = 0, than the test fails.</p><p>hint: if the top power is teh same or greater than the bottome power than series diverges!</p>
13
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Telescoping Series Test

factorable polynomial-do partial fractions and then usually BUT NOT ALWAYS have to break into two separate sigma notations

<p>factorable polynomial-do partial fractions and then usually BUT NOT ALWAYS have to break into two separate sigma notations</p>
14
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Absolute and Conditional Convergence

A series ∑an is called absolutely convergent if ∑|an|∑|an| is convergent. If ∑an is convergent and ∑|an| is divergent we call the series conditionally convergent.

<p>A series ∑an is called absolutely convergent if ∑|an|∑|an| is convergent. If ∑an is convergent and ∑|an| is divergent we call the series conditionally convergent.</p>
15
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Ratio Test for Absolute Convergence

Take the absolute value and do the ratio test.

do L=lim l an+1 / anl

n→∞

If lim < 1, absolutely converges

If lim > 1, diverges

If lim = 1, test fails

<p>Take the absolute value and do the ratio test.</p><p>do L=lim l an+1 / anl</p><p>n→∞</p><p>If lim &lt; 1, absolutely converges</p><p>If lim &gt; 1, diverges</p><p>If lim = 1, test fails</p>