Activity 8.3.6.
The series \(\sum \frac{1}{k^p}\) are special series called \(p\)-series. We have already seen that the \(p\)-series with \(p=1\) (the harmonic series) diverges. We investigate the behavior of other \(p\)-series in this activity.
(a)
Evaluate the improper integral \(\int_1^{\infty} \frac{1}{x^2} \ dx\text{.}\) Does the series \(\sum_{k=1}^{\infty} \frac{1}{k^2}\) converge or diverge? Explain.
(b)
Evaluate the improper integral \(\int_1^{\infty} \frac{1}{x^p} \ dx\) where \(p \gt 1\text{.}\) For which values of \(p\) can we conclude that the series \(\sum_{k=1}^{\infty} \frac{1}{k^p}\) converges?
(c)
Evaluate the improper integral \(\int_1^{\infty} \frac{1}{x^p} \ dx\) where \(p \lt 1\text{.}\) What does this tell us about the corresponding \(p\)-series \(\sum_{k=1}^{\infty} \frac{1}{k^p}\text{?}\)
(d)
Summarize your work in this activity by completing the following statement.
