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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
 test
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(Corollary)
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The following is an immediate corollary of the integral test.
Proof. The case  is well-known, for
 is the harmonic series, which diverges (see this proof). From now on, we assume  (notice that one could also use the integral test to prove the case  ). In order to apply the integral test, we need to calculate the following improper integral:
Since
 diverges when  and converges for  , the integral above converges for  , i.e. for  and diverges for  (and also diverges for  ). Therefore, the corollary follows by the integral test. 
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" test" is owned by alozano.
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(view preamble)
Cross-references: integral, improper integral, calculate, order, harmonic series, diverges, converges, series, integral test
There are 4 references to this entry.
This is version 3 of  test, born on 2005-03-21, modified 2008-04-01.
Object id is 6894, canonical name is PTest.
Accessed 5784 times total.
Classification:
| AMS MSC: | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
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Pending Errata and Addenda
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![$\displaystyle \int_1^\infty \frac{1}{x^p} dx=\lim_{n\to \infty}\left[ \frac{x^{1-p}}{1-p} \right]_1^n= \lim_{n\to\infty}\frac{n^{-p+1}}{1-p}-\frac{1}{1-p}.$](http://images.planetmath.org:8080/cache/objects/6894/l2h/img10.png "$\displaystyle \int_1^\infty \frac{1}{x^p} dx=\lim_{n\to \infty}\left[ \frac{x^{1-p}}{1-p} \right]_1^n= \lim_{n\to\infty}\frac{n^{-p+1}}{1-p}-\frac{1}{1-p}.$")