Difference: InfiniteSeriesSynopsis (6 vs. 7)

Revision 72008-11-16 - DickFurnas

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Infinite Series: A Synopsis

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Infinite Series: A Synopsis

 
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Famous Series

Geometric Series

  • if converges with sum
  • if diverges

P-Series

  • 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.

Exponential

  • converges

where

Sin

  • converges

Cos

  • 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

Geometric Series

P-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.

Exponential

where

Sin

Cos

ln (1+x)

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

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Useful Limits

 

Exponentials

Powers and Roots

Properties of Limits

The limit of a sum is the sum of the limits:

The limit of a product is the product of the limits:

The limit of a quotient is the quotient of the limits:

Squeeze Theorem:

AKA the "Flyswatter Theorem". Why?

smile

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

Always

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Useful Inequalities

Always

Eventually

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

New Series From Old

Multiply by a constant:

Before
After

Substitute an expression for x:

Before
After

Multiply by a power of x:

Before
After

Integrate:

Before
After

Differentiate:

Before
After
 
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Eventually

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

%INCLUDE{ConvergenceTests]]%
 
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Tips for Series

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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),
      so,
      which is a convergent p-series.

-- DickFurnas - 21 Oct 2008
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