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Revision 72008-11-16 - DickFurnas

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Infinite Series: A Synopsis

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Famous Series

Geometric Series

• if converges with sum
• if diverges

• for diverges

Harmonic Series

• diverges
• this is a special case of the P-Series for P=1

Alternating Harmonic Series

• converges to ln(1+1) = ln(2) using series for ln(1+x) below.

• converges

where

• converges

• converges

ln (1+x)

• you can arrive at this relation by integrating a Geometric Series in -t term-by-term.
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• if x=-1 , i.e. ln(1+(-1)) = ln(0) , this is the negative of the Harmonic Series which diverges toward -&infty;

arctan (x)

• you can arrive at this relation by integrating a Geometric Series in -t^2 term-by-term.
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Famous Series

Harmonic Series

• diverges
• this is a special case of the P-Series for P=1

Alternating Harmonic Series

• converges to ln(1+1) = ln(2) using series for ln(1+x) below.

where

ln (1+x)

• you can arrive at this relation by integrating a Geometric Series in -t term-by-term.
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• if x=-1 , i.e. ln(1+(-1)) = ln(0) , this is the negative of the Harmonic Series which diverges toward -∞

arctan (x)

• you can arrive at this relation by integrating a Geometric Series in -t2 term-by-term.
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Useful Limits

Ratios of Polynomials

The limiting behavior of the ratio of two polynomials depends on the degree of the polynomials. The polynomial of higher degree "wins". If their degrees are the same, then the limit is the ratio of the leading coefficients.

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Useful Inequalities

Eventually

1. for any
2. (!) Remember this when using the Test for Divergence

New Series From Old

Before
After

Before
After

Before
After

Before
After

Differentiate:

Before
After

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Eventually

1. for any
2. (!) Remember this when using the Test for Divergence
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Convergence Tests

%INCLUDE{ConvergenceTests]]%

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Tips for Series

• Often the hardest part of showing convergence or divergence of a series is the indecision: What do I believe it does? After all, you'll have a tough time showing a series converges if it doesn't!
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• The limits listed in another section can help a lot with the Test for Divergence. Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
• Many limits boil down to "look like" ratios of polynomials after stripping out trig functions using the Useful Inequalities for trig functions.
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• The limits listed in UsefulLimits can help a lot with the Test for Divergence. Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
• Many limits boil down to "look like" ratios of polynomials after stripping out trig functions using the Useful Inequalities for trig functions.

• The eventual behavior that for any leads to the peculiar rule of thumb that in lots of ratios ln(n) "looks like" 1 since any positive power of n will dominate it:
• informally, "looks like" so converges
• more carefully, (eventually),

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