Degree difference test: Difference between revisions
(Created page with "==Statement== ===For rational functions=== Consider a series of the form: <math>\sum_{k=k_0}^\infty \frac{p(k)}{q(k)}</math> where <math>p</math> and <math>q</math> are polyn...") |
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where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(k)</math> is nonzero for all integers <math>k \ge k_0</math>. Then the following rules hold: | where <math>p</math> and <math>q</math> are polynomials, <math>q</math> is not the zero polynomial, and <math>q(k)</math> is nonzero for all integers <math>k \ge k_0</math>. Then the following rules hold: | ||
# If <math>\deg q - \deg p \ge 2</math>, then the series is an [[absolutely convergent series]]. | # If <math>\! \deg q - \deg p \ge 2</math>, then the series is an [[absolutely convergent series]]. | ||
# If <math>\deg q - \deg p = 1</math>, then the series diverges, but the terms limit to zero. | # If <math>\! \deg q - \deg p = 1</math>, then the series diverges, but the terms limit to zero. | ||
# If <math>\deg p \ge deg q</math>, then the series diverges, and the terms do not limit to zero. | # If <math>\! \deg p \ge \deg q</math>, then the series diverges, and the terms do not limit to zero. | ||
Revision as of 20:35, 7 September 2011
Statement
For rational functions
Consider a series of the form:
where and are polynomials, is not the zero polynomial, and is nonzero for all integers . Then the following rules hold:
- If , then the series is an absolutely convergent series.
- If , then the series diverges, but the terms limit to zero.
- If , then the series diverges, and the terms do not limit to zero.
