Summary edit

DescriptionMercator series.png	English: Graph showing how the Mercator Series converges to ${\displaystyle \ln \left(x\right)}$ for ${\displaystyle 0<x\leq 2}$ with an increasing number of terms ${\displaystyle n}$ included in the finite series ${\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}(x-1)^{k}\,.}$ Dansk: Graf, som viser hvorledes ${\displaystyle n}$'te ordens Taylorpolynomiet ${\displaystyle P_{n}(x)=\sum _{k=1}^{n}{\frac {1}{k}}(x-1)^{k}\,,}$ for ${\displaystyle \ln(x)}$ i angrebspunktet ${\displaystyle x_{0}=1}$ konverger mod den naturlige logaritme for ${\displaystyle 0<x\leq 2}$ med et stigende antal led ${\displaystyle n}$ i den endelige række. I grænsen ${\displaystyle n\rightarrow \infty }$ er Taylorpolynomiet med substitutionen ${\displaystyle x\rightarrow 1+x}$identisk med Mercators række.
Date	18 January 2007
Source	Screen shot of own work using the GFDL program Graph.
Author	Kim Hansen

File:Mercator series.svg is a vector version of this file. It should be used in place of this PNG file. File:Mercator series.png → File:Mercator series.svg For more information, see Help:SVG. In other languages Alemannisch ∙ Bahasa Indonesia ∙ Bahasa Melayu ∙ British English ∙ català ∙ čeština ∙ dansk ∙ Deutsch ∙ eesti ∙ English ∙ español ∙ Esperanto ∙ euskara ∙ français ∙ Frysk ∙ galego ∙ hrvatski ∙ Ido ∙ italiano ∙ lietuvių ∙ magyar ∙ Nederlands ∙ norsk bokmål ∙ norsk nynorsk ∙ occitan ∙ Plattdüütsch ∙ polski ∙ português ∙ português do Brasil ∙ română ∙ Scots ∙ sicilianu ∙ slovenčina ∙ slovenščina ∙ suomi ∙ svenska ∙ Tiếng Việt ∙ Türkçe ∙ vèneto ∙ Ελληνικά ∙ беларуская (тарашкевіца) ∙ български ∙ македонски ∙ нохчийн ∙ русский ∙ српски / srpski ∙ татарча/tatarça ∙ українська ∙ ქართული ∙ հայերեն ∙ বাংলা ∙ தமிழ் ∙ മലയാളം ∙ ไทย ∙ 한국어 ∙ 日本語 ∙ 简体中文 ∙ 繁體中文 ∙ עברית ∙ العربية ∙ فارسی ∙ +/−

 

Instructions edit

The following Graph file can be used to generate the image.

;This file was created by Graph (http://www.padowan.dk)
;Do not change this file from other programs.
[Graph]
Version = 4.2.0.332
MinVersion = 2.5
OS = Windows XP 5.1 Service Pack 2

[Axes]
xMin = -1
xMax = 3
xTickUnit = 1
xGridUnit = 1
xShowGrid = 1
xAutoTick = 0
xAutoGrid = 0
yMin = -3
yMax = 3
yTickUnit = 1
yGridUnit = 1
yShowGrid = 1
yAutoTick = 0
yAutoGrid = 0
AxesColor = clBlack
GridColor = clMedGray
NumberFont = Times New Roman,20,clBlack
LabelFont = Times New Roman,20,clBlack,I
ShowLegend = 0
Radian = 1
LegendPlacement = 0
LegendPos = -0.694964028776978,1.76550680786687

[Func1]
FuncType = 0
y = ln(x)
LegendText = ln(x)
From = 0.001
To = 10
Steps = 1
Color = clRed
Size = 4

[Func2]
FuncType = 0
y = x-1
LegendText = n=1
From = -1
To = 10
Color = clMaroon
Size = 2

[Func3]
FuncType = 0
y = (x-1)-0.5*(x-1)^2
LegendText = n=2
From = -1
To = 10
Steps = 1
Color = clGreen
Size = 2

[Func4]
FuncType = 0
y = (x-1)+0.5*(x-1)^2+(x-1)^3/3
LegendText = n=3
Color = clOlive
Size = 2

[Func5]
FuncType = 0
y = (x-1)-0.5*(x-1)^2+(x-1)^3/3-(x-1)^4/4+(x-1)^5/5-(x-1)^6/6+(x-1)^7/7-(x-1)^8/8+(x-1)^9/9-(x-1)^10/10
LegendText = n=10
From = -1
To = 10
Color = clNavy
Size = 2

[Label1]
Placement = 0
Pos = 2.54421768707483,1.40916530278232
Text = {\\rtf1\\ansi\\ansicpg1252\\deff0\\deflang1030{\\fonttbl{\\f0\\fnil\\fprq5\\fcharset0 Times New Roman;}}\n{\\colortbl ;\\red255\\green0\\blue0;\\red0\\green0\\blue0;}\n\\viewkind4\\uc1\\pard\\cf1\\f0\\fs40 ln\\b (\\i x\\i0 )\\cf2\\b0\\fs24\\par\n}\n
BackgroundColor = clNone
ShowInLegend = 0

[Label2]
Placement = 0
Pos = 2.2312925170068,1.87070376432079
Text = {\\rtf1\\ansi\\ansicpg1252\\deff0\\deflang1030{\\fonttbl{\\f0\\fnil\\fcharset0 Times New Roman;}}\n{\\colortbl ;\\red128\\green0\\blue0;\\red255\\green0\\blue0;}\n\\viewkind4\\uc1\\pard\\cf1\\i\\f0\\fs40 n\\i0 =1\\cf2\\fs28\\par\n}\n
BackgroundColor = clNone
ShowInLegend = 0

[Label3]
Placement = 0
Pos = 2.56462585034014,0.613747954173486
Text = {\\rtf1\\ansi\\ansicpg1252\\deff0\\deflang1030{\\fonttbl{\\f0\\fnil\\fcharset0 Times New Roman;}}\n{\\colortbl ;\\red0\\green128\\blue0;\\red255\\green0\\blue0;}\n\\viewkind4\\uc1\\pard\\cf1\\i\\f0\\fs40 n\\i0 =2\\cf2\\fs28\\par\n}\n
BackgroundColor = clNone
ShowInLegend = 0

[Label4]
Placement = 0
Pos = 1.56462585034014,1.93944353518822
Text = {\\rtf1\\ansi\\ansicpg1252\\deff0\\deflang1030{\\fonttbl{\\f0\\fnil\\fcharset0 Times New Roman;}}\n{\\colortbl ;\\red128\\green128\\blue0;\\red0\\green128\\blue0;}\n\\viewkind4\\uc1\\pard\\cf1\\i\\f0\\fs40 n\\i0 =3\\cf2\\i\\fs28\\par\n}\n
BackgroundColor = clNone
ShowInLegend = 0

[Label5]
Placement = 0
Pos = 2.45578231292517,-0.682487725040917
Text = {\\rtf1\\ansi\\ansicpg1252\\deff0\\deflang1030{\\fonttbl{\\f0\\fnil\\fcharset0 Times New Roman;}}\n{\\colortbl ;\\red0\\green0\\blue128;\\red128\\green128\\blue0;}\n\\viewkind4\\uc1\\pard\\cf1\\i\\f0\\fs40 n\\i0 =10\\cf2\\fs28\\par\n}\n
BackgroundColor = clNone
ShowInLegend = 0

[Relation1]
Relation = x>2
Style = 4
Color = clGray
Size = 0

[Relation2]
Relation = x<0
Style = 4
Color = clGray
Size = 0

[Func6]
FuncType = 1
x = 2
y = t
From = -3
To = 3
Steps = 1000
Style = 1
Color = clGray

[Data]
TextLabelCount = 5
FuncCount = 6
PointSeriesCount = 0
ShadeCount = 0
RelationCount = 2
OleObjectCount = 0

Licensing edit

I, the copyright holder of this work, hereby publish it under the following licenses:

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
	You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
	This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue

This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic, 2.0 Generic and 1.0 Generic license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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