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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
examples using comparison test without limit
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(Example)
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Do the following series converge?
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(1) |
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(2) |
The general term of (1) may be estimated upwards: $$0 < \frac{1}{n^2+n+1} < \frac{1}{n^2+0+0} = \frac{1}{n^2}$$ Because $\sum_{n=1}^\infty \frac{1}{n^2}$ (an over-harmonic series) converges, then also (1) converges.
The general term of (2) may be estimated downwards: $$\frac{n^3+n+1}{n^4+n+1} > \frac{n^3+0+0}{n^4+n^4+n^4} = \frac{1}{3}\cdot\frac{1}{n} > 0$$ Because $\sum_{n=1}^\infty \frac{1}{3}\frac{1}{n}$ (the harmonic series divided by 3) diverges, then also (2) diverges.
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"examples using comparison test without limit" is owned by pahio.
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See Also:  test
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over-harmonic series |
This object's parent.
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Cross-references: diverges, harmonic series, converge, series
There are 3 references to this entry.
This is version 6 of examples using comparison test without limit, born on 2005-03-21, modified 2007-09-05.
Object id is 6895, canonical name is ExamplesUsingComparisonTestWithoutLimit.
Accessed 3424 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
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Pending Errata and Addenda
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