Difference: InfiniteSeriesSynopsis (1 vs. 7)

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.
<--/twistyPlugin twikiMakeVisibleInline-->

<--/twistyPlugin-->
• 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.
<--/twistyPlugin twikiMakeVisibleInline-->

>
>

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.
<--/twistyPlugin twikiMakeVisibleInline-->

<--/twistyPlugin-->
• 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.
<--/twistyPlugin twikiMakeVisibleInline-->

<--/twistyPlugin-->

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<
<

<--/twistyPlugin-->

>
>

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

Changed:
<
<

Eventually

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

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!
Changed:
<
<
• 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.
>
>
• 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),

Revision 62008-11-16 - DickFurnas

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 META TOPICPARENT name="MscCapsules"

Infinite Series: A Synopsis

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>
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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.

Useful Inequalities

Always

Revision 52008-10-30 - DickFurnas

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Infinite Series: A Synopsis

Revision 42008-10-29 - DickFurnas

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 META TOPICPARENT name="MscCapsules"

Infinite Series: A Synopsis

Line: 57 to 57

</>

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<
• if x=-1 ( ln(1+(-1)) = ln(0) ) this is the negative of the Harmonic Series which diverges toward -&infty;
>
>
• if x=-1 , i.e. ln(1+(-1)) = ln(0) , this is the negative of the Harmonic Series which diverges toward -&infty;

arctan (x)

Revision 32008-10-23 - DickFurnas

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 META TOPICPARENT name="MscCapsules"

Infinite Series: A Synopsis

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</>
<--/twistyPlugin-->

Useful Inequalities

Changed:
<
<
>
>

Changed:
<
<
>
>

Eventually

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

Changed:
<
<

"Eventually"

for any

(!) Remember this when using the Test for Divergence

>
>

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!
Changed:
<
<
• 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" 1/n and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
Error during latex2img:
ERROR: problems during latex
INPUT:
\documentclass[fleqn,12pt]{article}
\usepackage{amsmath}
\usepackage[normal]{xcolor}
\setlength{\mathindent}{0cm}
\definecolor{teal}{rgb}{0,0.5,0.5}
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\definecolor{silver}{gray}{0.75}
\usepackage{latexsym}
\begin{document}
\pagestyle{empty}
\pagecolor{white}
{
\color{black}
\begin{math}\displaystyle n \rarrow \infty\end{math}
}
\clearpage
\end{document}
STDERR:
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)
restricted \write18 enabled.
entering extended mode
(/tmp/jpLZVBAob5/Gn1IiM7_wl
LaTeX2e <2011/06/27>
Babel  and hyphenation patterns for english, dumylang, nohyphenation, lo
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
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(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the ?' option.
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(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
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(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/dvips.def))
(/usr/share/texlive/texmf-dist/tex/latex/base/latexsym.sty)
No file Gn1IiM7_wl.aux.
(/usr/share/texlive/texmf-dist/tex/latex/base/ulasy.fd)
! Undefined control sequence.
l.17 \begin{math}\displaystyle n \rarrow
\infty\end{math}
[1] (./Gn1IiM7_wl.aux) )
(see the transcript file for additional information)
Output written on Gn1IiM7_wl.dvi (1 page, 420 bytes).
Transcript written on Gn1IiM7_wl.log.

>
>
• 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.
• 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

Revision 22008-10-22 - DickFurnas

Line: 1 to 1

 META TOPICPARENT name="MscCapsules"

Infinite Series: A Synopsis

Line: 7 to 7

Famous Series

Geometric Series

Changed:
<
<
>
>

• if converges with sum
• if diverges
>
>

Changed:
<
<
>
>

• for diverges
>
>

Harmonic Series

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

Alternating Harmonic Series

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

• converges
>
>
where

Sin

• converges
Line: 38 to 43

>
>
• you can arrive at this relation by integrating a Geometric Series in -t term-by-term.
<--/twistyPlugin twikiMakeVisibleInline-->

<--/twistyPlugin-->
• if x=-1 ( 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.
<--/twistyPlugin twikiMakeVisibleInline-->

<--/twistyPlugin-->

Useful Inequalities

"Eventually"

for any

(!) Remember this when using the Test for Divergence

Tips

• 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!
• 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" 1/n and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
Error during latex2img:
ERROR: problems during latex
INPUT:
\documentclass[fleqn,12pt]{article}
\usepackage{amsmath}
\usepackage[normal]{xcolor}
\setlength{\mathindent}{0cm}
\definecolor{teal}{rgb}{0,0.5,0.5}
\definecolor{navy}{rgb}{0,0,0.5}
\definecolor{aqua}{rgb}{0,1,1}
\definecolor{lime}{rgb}{0,1,0}
\definecolor{maroon}{rgb}{0.5,0,0}
\definecolor{silver}{gray}{0.75}
\usepackage{latexsym}
\begin{document}
\pagestyle{empty}
\pagecolor{white}
{
\color{black}
\begin{math}\displaystyle n \rarrow \infty\end{math}
}
\clearpage
\end{document}
STDERR:
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)
restricted \write18 enabled.
entering extended mode
(/tmp/ElQ_mNh_Ng/QfRG3rKDwG
LaTeX2e <2011/06/27>
Babel  and hyphenation patterns for english, dumylang, nohyphenation, lo
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
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(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the ?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/dvips.def))
(/usr/share/texlive/texmf-dist/tex/latex/base/latexsym.sty)
No file QfRG3rKDwG.aux.
(/usr/share/texlive/texmf-dist/tex/latex/base/ulasy.fd)
! Undefined control sequence.
l.17 \begin{math}\displaystyle n \rarrow
\infty\end{math}
[1] (./QfRG3rKDwG.aux) )
(see the transcript file for additional information)
Output written on QfRG3rKDwG.dvi (1 page, 420 bytes).
Transcript written on QfRG3rKDwG.log.


-- DickFurnas - 21 Oct 2008

Revision 12008-10-21 - DickFurnas

Line: 1 to 1
>
>
 META TOPICPARENT name="MscCapsules"

Infinite Series: A Synopsis

Famous Series

Geometric Series

• if converges with sum
• if diverges

• for diverges

• diverges

• converges

• converges

• converges

• converges

arctan (x)

-- DickFurnas - 21 Oct 2008

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