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The R= atio Test and the Root test
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The R= atio Test and the Root test
= We shall assume all series are positive series, i.e. all terms in the series are positive unless stated <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>otherwise.
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One fact of the convergent geometric series 
The R= atio Test and the Root test
Σ
n=3D1
∞
rn
is that
an+1
an 
=3D
rn+1
rn 
=3D
r < 1.
lim
lim
nà∞
nà∞
= We shall assume all series are positive series, i.e. all terms in the series are positive unless stated <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>otherwise.
------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0056_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0056_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0050.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
In other words, the limit= of the ratio of the terms exists and is less than 1. 
One fact of the convergent geometric series 
The R= atio Test and the Root test
Σ
n=3D1
∞
rn
is that
an+1
an 
=3D
rn+1
rn 
=3D
r < 1.
lim
lim
nà∞
nà∞
= We shall assume all series are positive series, i.e. all terms in the series are positive unless stated <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>otherwise.
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In other words, the limit= of the ratio of the terms exists and is less than 1. 
One fact of the convergent geometric series 
The R= atio Test and the Root test
Σ
n=3D1
∞
rn
is that
an+1
an 
=3D
rn+1
rn 
=3D
r < 1.
lim
lim
nà∞
nà∞
= We shall assume all series are positive series, i.e. all terms in the series are positive unless stated <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>otherwise.
Theorem
(Ratio Test): 
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In other words, the limit= of the ratio of the terms exists and is less than 1. 
One fact of the convergent geometric series 
The R= atio Test and the Root test
Σ
n=3D1
∞
rn
is that
an+1
an 
=3D
rn+1
rn 
=3D
r < 1.
lim
lim
nà∞
nà∞
= We shall assume all series are positive series, i.e. all terms in the series are positive unless stated <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>otherwise.
Theorem
(Ratio Test): Given a series
an+1
an 
=3D
r < 1, then        converges.
lim
nà∞
Σ
n=3D1
∞
an
If  
Σ
n=3D1
∞
an
------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0052_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0052_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0052_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0053.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
In other words, the limit= of the ratio of the terms exists and is less than 1. 
One fact of the convergent geometric series 
The R= atio Test and the Root test
Σ
n=3D1
∞
rn
is that
an+1
an 
=3D
rn+1
rn 
=3D
r < 1.
lim
lim
nà∞
nà∞
= We shall assume all series are positive series, i.e. all terms in the series are positive unless stated <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>otherwise.
Theorem
(Ratio Test): Given a series
an+1
an 
=3D
r < 1, then        converges.
lim
nà∞
Σ
n=3D1
∞
an
If  
an+1
an 
=3D
r > 1, then        diverges.
lim
nà∞
Σ
n=3D1
∞
an
If  
Σ
n=3D1
∞
an
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In other words, the limit= of the ratio of the terms exists and is less than 1. 
One fact of the convergent geometric series 
The R= atio Test and the Root test
Σ
n=3D1
∞
rn
is that
an+1
an 
=3D
rn+1
rn 
=3D
r < 1.
lim
lim
nà∞
nà∞
= We shall assume all series are positive series, i.e. all terms in the series are positive unless stated <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>otherwise.
Theorem
(Ratio Test): Given a series
an+1
an 
=3D
r < 1, then        converges.
lim
nà∞
Σ
n=3D1
∞
an
If  
an+1
an 
=3D
r > 1, then        diverges.
lim
nà∞
Σ
n=3D1
∞
an
If  
an+1
an 
=3D
1, then the test is inconclusiv= e.
lim
nà∞
If  
Σ
n=3D1
∞
an
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The R= atio Test and the Root test
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
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The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
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The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
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The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
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The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
=3D
1
3
2n+1 + (n+1)3
2n + n3
[
]
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The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
=3D
1
3
2n+1 + (n+1)3
2n + n3
[
]
 (divide the top and bottom by 2n+1)
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The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
=3D
1
3
2n+1 + (n+1)3
2n + n3
[
]
 (divide the top and bottom by 2n+1)
=3D
1
3
1 + (n+1)3/2n+1
1/2 + n/2n+1
[
]
------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBK AAEAgAAAAAAAAAIHjI+py+3PCgA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABJ AAEAgAAAAAAAAAIHjI+py+2/CgA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAl AAEAgAAAAAAAAAIFjI+py1EAO1== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0061_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQB/ AAEAgAAAAAAAAAIKjI+py+0Po5yrAAA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
=3D
1
3
2n+1 + (n+1)3
2n + n3
[
]
 (divide the top and bottom by 2n+1)
=3D
1
3
1 + (n+1)3/2n+1
1/2 + n/2n+1
[
]
as nà0,
0
------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBK AAEAgAAAAAAAAAIHjI+py+3PCgA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABJ AAEAgAAAAAAAAAIHjI+py+2/CgA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAl AAEAgAAAAAAAAAIFjI+py1EAO1== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQB/ AAEAgAAAAAAAAAIKjI+py+0Po5yrAAA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0062_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhjAAdAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACJ ABwAgAAAAAAAAAKXjI+py+0PIwS02oszDrr7D4bi2HHkaZ7qyrZW6pbxTNcVbN/5zov4/usJhUFb cYisHWnLpJPVjEWfVF/O8KpqXdMRgtLdirNSx/hs9UpMYfS2KNlgwe66BmZete1P/P7zx4cU6CVo SJhmaIcYwqhoNPi4GCnp5ghYaUmZOXbp4cmptxmqBXpHKmaaoYraOLgGGys7C5tUAAA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
=3D
1
3
2n+1 + (n+1)3
2n + n3
[
]
 (divide the top and bottom by 2n+1)
=3D
1
3
1 + (n+1)3/2n+1
1/2 + n/2n+1
[
]
as nà0,
0
0
------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBK AAEAgAAAAAAAAAIHjI+py+3PCgA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABJ AAEAgAAAAAAAAAIHjI+py+2/CgA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image059.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAl AAEAgAAAAAAAAAIFjI+py1EAO1== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image060.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBy AAEAgAAAAAAAAAIJjI+py+0Po2wFADs= ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image061.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image062.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQB/ AAEAgAAAAAAAAAIKjI+py+0Po5yrAAA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0063_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhjAA2AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACJ ADMAgAAAAAAAAALxjI+py+0PIwS02oszDrr7D4bi2HHkaZ7qyrZW6pbxTNcVbN/5zov4/usJhUFb cYisHWnLpJPVjEWfVF/O8KpqXdMRgtLdirNSx/hs9UpMYfS2KNlgwe66BmZete1P/P7zx4cU6CVo SJhmaIcYwqhoNPi4GCnp5ghYaUmZOXbp4cmptxmqBXpHKmaaoYraOLgGGys7C9t6wUpkC4Lbw9vZ 6psKrIuJGvw2TCxjrLxMelya3LwhPU32bE3NnH0d6geduw2WQAzOdEeOLY7+lWk+E6bw+C6lIn/7 u26fPhetD7UA2b8yB0rROngQSDpu6j4UAAA7 ------=_NextPart_01C97CE6.24D6A9A0 Content-Location: file:///C:/AA7BA2D4/26TheRatioandRootTest_files/slide0064.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
=3D
1
3
2n+1 + (n+1)3
2n + n3
[
]
 (divide the top and bottom by 2n+1)
=3D
1
3
1 + (n+1)3/2n+1
1/2 + n/2n+1
[
]
as nà0, we get the limit
0
0
2
3 .
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The R= atio Test and the Root test
Example:
Let an =3D
3n
2n + n3
When we use the ratio test, it is easier = to check
an+1* 1/an directly.
, then an+1 =3D
3n+1
2n+1 + (n+1)= 3
So an+1 * 1/an =3D
3n+1
2n+1 + (n+1)3
3n
2n + n3
=3D
1
3
2n+1 + (n+1)3
2n + n3
[
]
 (divide the top and bottom by 2n+1)
=3D
1
3
1 + (n+1)3/2n+1
1/2 + n/2n+1
[
]
as nà0, we get the limit
0
0
2
3 .
Hence the<= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;color:#006600;mso-fareast-language:ZH-CN'> series        = ;            converges.
Σ
n=3D1
∞
3n
2n + n3
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