## 17Calculus Proofs - p-Series Proof

##### 17Calculus

This page contains a proof of the p-series convergence theorem. Using this theorem and practice problems are on a separate page.

p-Series Convergence Theorem

The p-series $$\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^p}}}$$

To prove this, we will use the Integral Test and the Special Improper Integral.

The integral test tells us that we can set up the integral $$\displaystyle{ \int_1^{\infty}{\frac{1}{x^p} dx} }$$

If this integral converges, then so does the series. Similarly, if the integral diverges, the series also diverges.

We will look at these five cases.
1. $$p > 1$$
2. $$p = 1$$
3. $$0 \lt p \leq 1$$
4. $$p = 0$$
5. $$p \lt 0$$
Note: Although the last two cases are not part of the theorem, we will show what happens in those two cases to answer the question at the top of the page and for completeness.

Let's use the Integral Test to prove convergence and divergence by calculating the corresponding improper integrals. For most of the cases, we need to set up a limit as follows

$$\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x^p} dx}} }$$

Since the function $$\displaystyle{ \frac{1}{x^p} }$$ is continuous on the interval $$[1,b]$$, this integral can be evaluated. So, now let's look at each case individually.

Case 1: $$p > 1$$
When $$p > 1$$, the integral is continuous and decreasing on the interval. So the Integral Test applies.

 $$\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x^p} dx}} }$$ $$\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{ x^{-p} dx}} }$$ $$\displaystyle{ \lim_{b \to \infty}{ \left[ \frac{x^{-p+1}}{-p+1} \right]_{1}^{b} } }$$ $$\displaystyle{ \frac{1}{-p+1} \lim_{b \to \infty}{\left[ b^{-p+1} - 1^{-p+1} \right]} }$$ $$\displaystyle{ \frac{1}{-p+1} [ 0 - 1 ] = \frac{1}{p-1} }$$

Since $$\displaystyle{ \frac{1}{p-1} }$$ is finite, the integral converges and, therefore, by the Integral Test, the series also converges.

Case 2: $$p=1$$
When $$p = 1$$, we have

$$\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x} dx}} \to \lim_{b \to \infty }{ \left[ \ln(x) \right]_{1}^{b} } \to \lim_{b \to \infty }{ \ln(b) } - 0 = \infty }$$
Since the limit is infinity, the series diverges.

Case 3: $$0 \lt p \lt 1$$
We will use the Integral Test again.

 $$\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x^p} dx}} }$$ $$\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{ x^{-p} dx}} }$$ $$\displaystyle{ \lim_{b \to \infty}{ \left[ \frac{x^{-p+1}}{-p+1} \right]_{1}^{b} } }$$ $$\displaystyle{ \frac{1}{-p+1} \lim_{b \to \infty}{\left[ b^{-p+1} - 1^{-p+1} \right]} }$$

So far in these calculations, we have the same equation as we did in case 1. However, in this case, $$0 \lt p \lt 1$$, which means that the exponent $$-p+1 > 0$$ and therefore $$\displaystyle{ \lim_{b \to \infty}{ b^{-p+1} } \to \infty }$$. So the entire integral diverges, which means that the series diverges.

Case 4: $$p = 0$$
When $$p = 0$$, the series is $$\displaystyle{ \sum_{n=1}^{\infty}{1} }$$. Since $$\displaystyle{ \lim_{n \to \infty}{1} = 1 \neq 0 }$$ the divergence test tells us that the series diverges.

Case 5: $$p \lt 0$$
We can write the fraction $$\displaystyle{ \frac{1}{n^p} = n^{-p} }$$. Since $$p \lt 0$$, the exponent here is positive and so the terms are increasing. Since $$\displaystyle{ \lim_{n \to \infty}{ \frac{1}{n^p} } \neq 0 }$$, the series diverges by the divergence test.

In Summary: For the series $$\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^p}}}$$
1. $$p > 1$$ converges by the integral test
2. $$p = 1$$ diverges by the integral test
3. $$0 \lt p \lt 1$$ diverges by the integral test
4. $$p = 0$$ diverges by the divergence test
5. $$p \lt 0$$ diverges by the divergence test

- qed -

Note: We have also seen the p-series theorem where, instead of writing $$0 \lt p \leq 1$$ for where the series diverges, it says the series diverges for $$p \leq 1$$. Of course, this includes all of the cases above. This is a more efficient way to write the theorem statement.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.