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The Integral Test + Example on Convergence of a Series - Hive - YouTube - Notes - Playlist - Sequences and Series In this video, I go over the Integral Test, which states that a series converges only if its corresponding improper integral also converges.…
The p-Series ∑ 1/n^p is convergent if p is greater than 1 - Hive - YouTube - Notes - Playlist - Sequences and Series
In this video, I use the Integral Test to show that the p-series, whose terms are 1/n^p, is convergent if p is greater than 1 and divergent for all other values of p. When p is less than 0, the terms approach infinity; when p = 0, the terms equal 1, hence it is divergent by the test for divergence; and when p = 1, we get the harmonic series, which is also divergent. I also illustrate this with examples on a convergent and a divergent p-series. Note also that when using the Integral Test, the corresponding improper integrals do NOT, in general, equal the sum of the series.
#math #calculus #series #pseries #education
In this video, I use the Integral Test to show that the p-series, whose terms are 1/n^p, is convergent if p is greater than 1 and divergent for all other values of p. When p is less than 0, the terms approach infinity; when p = 0, the terms equal 1, hence it is divergent by the test for divergence; and when p = 1, we get the harmonic series, which is also divergent. I also illustrate this with examples on a convergent and a divergent p-series. Note also that when using the Integral Test, the corresponding improper integrals do NOT, in general, equal the sum of the series.
#math #calculus #series #pseries #education
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