Strategies for Testing Series - Revision history

Kayla Murray at 18:50, 30 October 2017

2017-10-30T18:50:22Z

Kayla Murray at 18:46, 30 October 2017

2017-10-30T18:46:49Z

Kayla Murray at 20:01, 22 October 2017

2017-10-22T20:01:58Z

Kayla Murray at 19:52, 22 October 2017

2017-10-22T19:52:32Z

Kayla Murray at 19:46, 22 October 2017

2017-10-22T19:46:42Z

Kayla Murray at 19:33, 22 October 2017

2017-10-22T19:33:10Z

Kayla Murray at 19:09, 22 October 2017

2017-10-22T19:09:39Z

Kayla Murray: Created page with "text"

2017-10-22T19:03:32Z

Created page with "text"

New page

text

@@ Line 39: / Line 39: @@
 :with &nbsp;<math style="vertical-align: -4px">b_n>0</math>&nbsp; for all &nbsp;<math style="vertical-align: -4px">n,</math>&nbsp; then the Alternating Series Test should be considered.
-'''5.''' If the series involves factorials or other products,
+'''5.''' If the series involves factorials or other products, the Ratio Test should be considered.
 :<u>NOTE:</u> The Ratio Test should not be used for rational functions of &nbsp;<math style="vertical-align: 0px">n.</math>&nbsp;
@@ Line 53: / Line 51: @@
 :is easily evaluated, the Integral Test should be considered.
-'''7.''' If all of the terms in the series have powers involving &nbsp;<math style="vertical-align: -4px">n,</math>&nbsp;
+'''7.''' If the terms of the series are products involving powers of &nbsp;<math style="vertical-align: -4px">n,</math>&nbsp;
 :then the Root Test should be considered.

@@ Line 7: / Line 7: @@
 '''1.''' If the series is of the form
-::<math style="vertical-align: -10px">\sum \frac{1}{n^p} </math>&nbsp; or &nbsp;<math style="vertical-align: -5px">\sum ar^n,</math>&nbsp;
+:::<math style="vertical-align: -10px">\sum \frac{1}{n^p} </math>&nbsp; or &nbsp;<math style="vertical-align: -5px">\sum ar^n,</math>&nbsp;
-then the series is a &nbsp;<math style="vertical-align: -4px">p-</math>series or a geometric series
+:then the series is a &nbsp;<math style="vertical-align: -4px">p-</math>series or a geometric series
-For the &nbsp;<math style="vertical-align: -4px">p-</math>series
+:For the &nbsp;<math style="vertical-align: -4px">p-</math>series
-::<math>\sum \frac{1}{n^p},</math>&nbsp;
+:::<math>\sum \frac{1}{n^p},</math>&nbsp;
-it is convergent if &nbsp;<math style="vertical-align: -4px">p>1</math>&nbsp; and divergent if &nbsp;<math style="vertical-align: -4px">p\le 1.</math>
+:it is convergent if &nbsp;<math style="vertical-align: -4px">p>1</math>&nbsp; and divergent if &nbsp;<math style="vertical-align: -4px">p\le 1.</math>
-For the geometric series
+:For the geometric series
-::<math>\sum ar^n,</math>&nbsp;
+:::<math>\sum ar^n,</math>&nbsp;
-it is convergent if &nbsp;<math style="vertical-align: -5px">|r|<1</math>&nbsp; and divergent if &nbsp;<math style="vertical-align: -4px">|r|\ge 1.</math>
+:it is convergent if &nbsp;<math style="vertical-align: -5px">|r|<1</math>&nbsp; and divergent if &nbsp;<math style="vertical-align: -4px">|r|\ge 1.</math>
-'''2.''' If the series has a form similar to a &nbsp;<math style="vertical-align: -4px">p-</math>series or a geometric series, then one of the comparison tests should be considered.
+'''2.''' If the series has a form similar to a &nbsp;<math style="vertical-align: -4px">p-</math>series or a geometric series,
 '''3.''' If you can see that
-::<math>\lim_{n\rightarrow \infty} a_n \neq 0,</math>&nbsp;
+:::<math>\lim_{n\rightarrow \infty} a_n \neq 0,</math>&nbsp;
-then you should use the Divergence Test or &nbsp;<math style="vertical-align: 0px">n</math>th term test.
+:then you should use the Divergence Test or &nbsp;<math style="vertical-align: 0px">n</math>th term test.
 '''4.''' If the series has the form
-::<math style="vertical-align: -6px">\sum (-1)^n b_n</math>&nbsp; or &nbsp;<math style="vertical-align: -6px">\sum (-1)^{n-1} b_n</math>&nbsp;
+:::<math style="vertical-align: -6px">\sum (-1)^n b_n</math>&nbsp; or &nbsp;<math style="vertical-align: -6px">\sum (-1)^{n-1} b_n</math>&nbsp;
-with &nbsp;<math style="vertical-align: -4px">b_n>0</math>&nbsp; for all &nbsp;<math style="vertical-align: -4px">n,</math>&nbsp; then the Alternating Series Test should be considered.
+:with &nbsp;<math style="vertical-align: -4px">b_n>0</math>&nbsp; for all &nbsp;<math style="vertical-align: -4px">n,</math>&nbsp; then the Alternating Series Test should be considered.
-'''5.''' If the series involves factorials or other products (including constants raised to the &nbsp;<math style="vertical-align: 0px">n</math>th power), the Ratio Test should be considered.
+'''5.''' If the series involves factorials or other products,
-<u>NOTE:</u> The Ratio Test should not be used for rational functions of &nbsp;<math style="vertical-align: 0px">n.</math>
+:the Ratio Test should be considered.
 '''6.''' If &nbsp;<math style="vertical-align: -5px">a_n=f(n)</math>&nbsp; for some function &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; where
-::<math>\int_a^\infty f(x)~dx</math>&nbsp;
+:::<math>\int_a^\infty f(x)~dx</math>&nbsp;
-is easily evaluated, the Integral Test should be considered (if all the hypothesis of the Integral Test are satisfied).
+:then the Root Test should be considered.
 <u>NOTE:</u> These strategies are used for determining whether a series converges or diverges.

@@ Line 7: / Line 7: @@
 '''1.''' If the series is of the form
-::<math>\sum \frac{1}{n^p} </math>&nbsp; or &nbsp;<math>\sum ar^n,</math>&nbsp;
+::<math style="vertical-align: -10px">\sum \frac{1}{n^p} </math>&nbsp; or &nbsp;<math style="vertical-align: -5px">\sum ar^n,</math>&nbsp;
-then the series is a &nbsp;<math>p</math>-series or a geometric series
+then the series is a &nbsp;<math style="vertical-align: -4px">p-</math>series or a geometric series
-For the &nbsp;<math>p</math>-series
+For the &nbsp;<math style="vertical-align: -4px">p-</math>series
 ::<math>\sum \frac{1}{n^p},</math>&nbsp;
-it is convergent if &nbsp;<math>p>1</math>&nbsp; and divergent if &nbsp;<math>p\le 1.</math>
+it is convergent if &nbsp;<math style="vertical-align: -4px">p>1</math>&nbsp; and divergent if &nbsp;<math style="vertical-align: -4px">p\le 1.</math>
 For the geometric series
@@ Line 21: / Line 21: @@
 ::<math>\sum ar^n,</math>&nbsp;
-it is convergent if &nbsp;<math>|r|<1</math>&nbsp; and divergent if &nbsp;<math>|r|\ge 1.</math>
+it is convergent if &nbsp;<math style="vertical-align: -5px">|r|<1</math>&nbsp; and divergent if &nbsp;<math style="vertical-align: -4px">|r|\ge 1.</math>
-'''2.''' If the series has a form similar to a &nbsp;<math>p</math>-series or a geometric series, then one of the comparison tests should be considered.
+'''2.''' If the series has a form similar to a &nbsp;<math style="vertical-align: -4px">p-</math>series or a geometric series, then one of the comparison tests should be considered.
 '''3.''' If you can see that
@@ Line 29: / Line 29: @@
 ::<math>\lim_{n\rightarrow \infty} a_n \neq 0,</math>&nbsp;
-then you should use the Divergence Test or &nbsp;<math>n</math>th term test.
+then you should use the Divergence Test or &nbsp;<math style="vertical-align: 0px">n</math>th term test.
 '''4.''' If the series has the form
-::<math>\sum (-1)^n b_n</math>&nbsp; or &nbsp;<math>\sum (-1)^{n-1} b_n</math>&nbsp;
+::<math style="vertical-align: -6px">\sum (-1)^n b_n</math>&nbsp; or &nbsp;<math style="vertical-align: -6px">\sum (-1)^{n-1} b_n</math>&nbsp;
-with &nbsp;<math>b_n>0</math>&nbsp; for all &nbsp;<math>n,</math>&nbsp; then the Alternating Series Test should be considered.
+with &nbsp;<math style="vertical-align: -4px">b_n>0</math>&nbsp; for all &nbsp;<math style="vertical-align: -4px">n,</math>&nbsp; then the Alternating Series Test should be considered.
-'''5.''' If the series involves factorials or other products (including constants raised to the &nbsp;<math>n</math>th power), the Ratio Test should be considered.
+'''5.''' If the series involves factorials or other products (including constants raised to the &nbsp;<math style="vertical-align: 0px">n</math>th power), the Ratio Test should be considered.
-<u>NOTE:</u> The Ratio Test should not be used for rational functions of &nbsp;<math>n.</math>
+<u>NOTE:</u> The Ratio Test should not be used for rational functions of &nbsp;<math style="vertical-align: 0px">n.</math>
-'''6.''' If &nbsp;<math>a_n=f(n)</math>&nbsp; for some function &nbsp;<math>f(x)</math>&nbsp; where
+'''6.''' If &nbsp;<math style="vertical-align: -5px">a_n=f(n)</math>&nbsp; for some function &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; where
 ::<math>\int_a^\infty f(x)~dx</math>&nbsp;

@@ Line 5: / Line 5: @@
 This list is meant to serve as a guideline for which tests you should consider applying to a given series.
-'''1.''' If the series is of the form &nbsp;<math>\sum \frac{1}{n^p} </math>&nbsp; or &nbsp;<math>\sum ar^n,</math>&nbsp; then the series is a &nbsp;<math>p</math>-series or a geometric series
+'''1.''' If the series is of the form
-For the &nbsp;<math>p</math>-series &nbsp;<math>\sum \frac{1}{n^p},</math>&nbsp; it is convergent if &nbsp;<math>p>1</math>&nbsp; and divergent if &nbsp;<math>p\le 1.</math>
+::<math>\sum \frac{1}{n^p} </math>&nbsp; or &nbsp;<math>\sum ar^n,</math>&nbsp;
-For the geometric series &nbsp;<math>\sum ar^n,</math>&nbsp; it is convergent if &nbsp;<math>|r|<1</math>&nbsp; and divergent if &nbsp;<math>|r|\ge 1.</math>
+then the series is a &nbsp;<math>p</math>-series or a geometric series
 '''2.''' If the series has a form similar to a &nbsp;<math>p</math>-series or a geometric series, then one of the comparison tests should be considered.
-'''3.''' If you can see that &nbsp;<math>\lim_{n\rightarrow \infty} a_n \neq 0,</math> then you should use the Divergence Test or &nbsp;<math>n</math>th term test.
+'''3.''' If you can see that
-'''4.''' If the series has the form &nbsp;<math>\sum (-1)^n b_n</math>&nbsp; or &nbsp;<math>\sum (-1)^{n-1} b_n</math>&nbsp; with &nbsp;<math>b_n>0</math>&nbsp; for all &nbsp;<math>n,</math>&nbsp; then the Alternating Series Test should be considered.
+is easily evaluated, the Integral Test should be considered (if all the hypothesis of the Integral Test are satisfied).
-'''5.''' If the series involves factorials or other products (including constants raised to the &nbsp;<math>n</math>th power), the Ratio Test should be considered. '''NOTE:''' The Ratio Test should not be used for rational functions of &nbsp;<math>n.</math>
+<u>NOTE:</u> These strategies are used for determining whether a series converges or diverges.
-'''6.''' If &nbsp;<math>a_n=f(n)</math>&nbsp; for some function &nbsp;<math>f(x)</math>&nbsp; where &nbsp;<math>\int_a^\infty f(x)~dx</math>&nbsp; is easily evaluated, the Integral Test should be considered (if all the hypothesis of the Integral Test are satisfied).
+However, these are not the strategies one should use if we are determining whether or not a
-'''NOTE:''' These strategies are used for determining whether a series converges or diverges. However, these are not the strategies one should use if we are determining whether or not a series is absolutely convergent.
+series is absolutely convergent.

@@ Line 3: / Line 3: @@
 Instead, we classify series by their form and give tips as to which tests should be considered.
-This list is just meant to serve as a guideline for which tests you should consider applying to a given series.
+This list is meant to serve as a guideline for which tests you should consider applying to a given series.
 '''1.''' If the series is of the form &nbsp;<math>\sum \frac{1}{n^p} </math>&nbsp; or &nbsp;<math>\sum ar^n,</math>&nbsp; then the series is a &nbsp;<math>p</math>-series or a geometric series

← Older revision		Revision as of 19:09, 22 October 2017
Line 1:		Line 1:
−	~~text~~	+	In general, there are no specific rules as to which test to apply to a given series.
		+
		+	Instead, we classify series by their form and give tips as to which tests should be considered.
		+
		+	This list is just meant to serve as a guideline for which tests you should consider applying to a given series.
		+
		+	'''1.'''
		+
		+	'''2.'''
		+
		+	'''3.'''
		+
		+	'''4.'''
		+
		+	'''5.'''
		+
		+	'''6.'''
		+
		+	'''NOTE:'''