Difference between revisions of "Strategies for Testing Series"
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'''1.''' If the series is of the form | '''1.''' If the series is of the form | ||
| − | ::<math>\sum \frac{1}{n^p} </math> or <math>\sum ar^n,</math> | + | ::<math style="vertical-align: -10px">\sum \frac{1}{n^p} </math> or <math style="vertical-align: -5px">\sum ar^n,</math> |
| − | then the series is a <math>p</math> | + | then the series is a <math style="vertical-align: -4px">p-</math>series or a geometric series |
| − | For the <math>p</math> | + | For the <math style="vertical-align: -4px">p-</math>series |
::<math>\sum \frac{1}{n^p},</math> | ::<math>\sum \frac{1}{n^p},</math> | ||
| − | it is convergent if <math>p>1</math> and divergent if <math>p\le 1.</math> | + | it is convergent if <math style="vertical-align: -4px">p>1</math> and divergent if <math style="vertical-align: -4px">p\le 1.</math> |
For the geometric series | For the geometric series | ||
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::<math>\sum ar^n,</math> | ::<math>\sum ar^n,</math> | ||
| − | it is convergent if <math>|r|<1</math> and divergent if <math>|r|\ge 1.</math> | + | it is convergent if <math style="vertical-align: -5px">|r|<1</math> and divergent if <math style="vertical-align: -4px">|r|\ge 1.</math> |
| − | '''2.''' If the series has a form similar to a <math>p</math> | + | '''2.''' If the series has a form similar to a <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 | '''3.''' If you can see that | ||
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::<math>\lim_{n\rightarrow \infty} a_n \neq 0,</math> | ::<math>\lim_{n\rightarrow \infty} a_n \neq 0,</math> | ||
| − | then you should use the Divergence Test or <math>n</math>th term test. | + | then you should use the Divergence Test or <math style="vertical-align: 0px">n</math>th term test. |
'''4.''' If the series has the form | '''4.''' If the series has the form | ||
| − | ::<math>\sum (-1)^n b_n</math> or <math>\sum (-1)^{n-1} b_n</math> | + | ::<math style="vertical-align: -6px">\sum (-1)^n b_n</math> or <math style="vertical-align: -6px">\sum (-1)^{n-1} b_n</math> |
| − | with <math>b_n>0</math> for all <math>n,</math> then the Alternating Series Test should be considered. | + | with <math style="vertical-align: -4px">b_n>0</math> for all <math style="vertical-align: -4px">n,</math> then the Alternating Series Test should be considered. |
| − | '''5.''' If the series involves factorials or other products (including constants raised to the <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 <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 <math>n.</math> | + | <u>NOTE:</u> The Ratio Test should not be used for rational functions of <math style="vertical-align: 0px">n.</math> |
| − | '''6.''' If <math>a_n=f(n)</math> for some function <math>f(x)</math> where | + | '''6.''' If <math style="vertical-align: -5px">a_n=f(n)</math> for some function <math style="vertical-align: -5px">f(x)</math> where |
::<math>\int_a^\infty f(x)~dx</math> | ::<math>\int_a^\infty f(x)~dx</math> | ||
Revision as of 12:01, 22 October 2017
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 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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \frac{1}{n^p} } or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum ar^n,}
then the series is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p-} series or a geometric series
For the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p-} series
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \frac{1}{n^p},}
it is convergent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p>1} and divergent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\le 1.}
For the geometric series
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum ar^n,}
it is convergent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|<1} and divergent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |r|\ge 1.}
2. If the series has a form similar to a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p-} series or a geometric series, then one of the comparison tests should be considered.
3. If you can see that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow \infty} a_n \neq 0,}
then you should use the Divergence Test or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th term test.
4. If the series has the form
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum (-1)^n b_n} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum (-1)^{n-1} b_n}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n>0} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n,} then the Alternating Series Test should be considered.
5. If the series involves factorials or other products (including constants raised to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th power), the Ratio Test should be considered.
NOTE: The Ratio Test should not be used for rational functions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n.}
6. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n=f(n)} for some function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} where
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty f(x)~dx}
is easily evaluated, the Integral Test should be considered (if all the hypothesis of the Integral Test are satisfied).
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.