Help with a subtle point concerning the proof of the p-series test

In summary, the conversation discusses the use of the integral test to prove the p-series test, specifically focusing on the case when p≠1. The integral is shown to diverge or converge depending on the value of (-p+1), but the question arises about what happens when (-p+1) approaches 0. The conclusion is that in the p-series convergence theorem, p remains fixed and the limit will not approach 1.
  • #1
JJHK
24
1

Homework Statement



Hi, I'm trying to prove the p-series test, using the integral test. Everything seems to work out fine until I get into this point:

Consider when p≠1 (p=1 is easy to see that it diverges, so I will ignore that one).

Then we have:

∫(n)^(-p) dn = [ 1 / (-p+1) ] * n^(-p+1)



For very large (-p+1) or very negatively large (-p+1), it is clear to see that the integral diverges and converges, respectively. But what if (-p+1) → 0? How do we know that the integral converges if p>1 and diverges if p<1? Shouldn't the limit approach 1 as anything to the zeroth power is equal to 1?
 
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  • #2
In the p-series convergence theorem, p is fixed. If it is a bit different from 1, it will stay a bit different from 1. You need not worry about this.
 

1. What is the p-series test used for?

The p-series test is used to determine whether a series (a sum of infinitely many terms) converges or diverges. It is specifically used for series with terms in the form of 1/n^p, where n is the index of the term and p is a positive constant.

2. What is the subtle point in the proof of the p-series test?

The subtle point in the proof of the p-series test is the comparison of the given series with a corresponding integral.

3. How does the comparison with an integral help in proving convergence or divergence?

The comparison with an integral helps in proving convergence or divergence by using the fact that the sum of a series can be approximated by the area under the corresponding curve of the integrand.

4. Can the p-series test be used for series with negative or non-integer exponents?

No, the p-series test can only be used for series with positive integer exponents. For series with negative or non-integer exponents, other convergence tests such as the limit comparison test or the ratio test must be used.

5. Are there any exceptions to the p-series test?

Yes, there are a few exceptions to the p-series test. For example, the series 1/n^p does not converge when p = 1, as it becomes the harmonic series which is known to diverge. In addition, the p-series test cannot be used for series with alternating signs, as it only applies to series with positive terms.

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