A p-series is an infinite series of the form 1/1^p + 1/2^p + 1/3^p + ..., where p is a constant exponent. It is one of the most important benchmark series in calculus because it gives a simple rule for convergence. Many other series can be compared to a p-series to decide whether they converge or diverge.
The key idea is that the terms must shrink fast enough for the infinite sum to approach a finite value.
The convergence rule for a p-series comes from the integral test. The graph of f(x) = 1/x^p is positive, continuous, and decreasing for x >= 1 when p > 0, so the infinite series can be compared with the improper integral from 1 to infinity. If p > 1, the integral has a finite value and the series converges.
If p <= 1, the integral diverges and the series diverges.
Key Facts
- A p-series has the form sum from n = 1 to infinity of 1/n^p.
- The p-series sum 1/n^p converges if p > 1.
- The p-series sum 1/n^p diverges if p <= 1.
- The harmonic series is the p-series with p = 1: sum 1/n.
- Integral test link: integral from 1 to infinity of 1/x^p dx converges if p > 1.
- For p > 1, integral from 1 to infinity of 1/x^p dx = 1/(p - 1).
Vocabulary
- p-Series
- An infinite series of the form sum from n = 1 to infinity of 1/n^p, where p is a constant.
- Convergence
- A series converges when its sequence of partial sums approaches a finite number.
- Divergence
- A series diverges when its partial sums do not approach a finite number.
- Integral Test
- A test that compares an infinite series with an improper integral of a related positive, continuous, decreasing function.
- Harmonic Series
- The divergent p-series sum from n = 1 to infinity of 1/n, which occurs when p = 1.
Common Mistakes to Avoid
- Thinking any series with terms going to 0 converges. This is wrong because terms approaching 0 is necessary but not sufficient, as shown by the harmonic series.
- Using the wrong cutoff p >= 1 for convergence. The p-series converges only when p > 1, while p = 1 diverges.
- Forgetting that p can be less than or equal to 0. If p <= 0, the terms do not approach 0, so the series must diverge.
- Applying the integral test without checking the function conditions. The function should be positive, continuous, and decreasing on the interval being used.
Practice Questions
- 1 Determine whether the series sum from n = 1 to infinity of 1/n^3 converges or diverges.
- 2 Determine whether the series sum from n = 1 to infinity of 1/sqrt(n) converges or diverges, and identify the value of p.
- 3 Explain why sum from n = 1 to infinity of 1/n diverges even though the terms 1/n approach 0.