p-series for Master Mathematics
What is a P-Series?
A p-series is a series of the form:
$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$
where $p$ is a constant. These are benchmark series — you'll compare other series to them constantly.
The Convergence Rule
$$\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges if } p > 1, \text{ diverges if } p \leq 1$$
That's it! Just check if $p > 1$.
Why Does This Work?
The intuition: How fast do the terms shrink?
- If $p$ is large, terms like $\frac{1}{n^p}$ shrink very fast → sum stays finite
- If $p$ is small, terms shrink slowly (or not at all) → sum blows up
The boundary is $p = 1$ (the harmonic series), which diverges just barely.
Key Examples to Memorize
| Series | Value of $p$ | Converges? |
|---|---|---|
| $\displaystyle\sum \frac{1}{n^2}$ | $p = 2$ | ✓ Yes |
| $\displaystyle\sum \frac{1}{n^3}$ | $p = 3$ | ✓ Yes |
| $\displaystyle\sum \frac{1}{\sqrt{n}} = \sum \frac{1}{n^{1/2}}$ | $p = \frac{1}{2}$ | ✗ No |
| $\displaystyle\sum \frac{1}{n}$ | $p = 1$ | ✗ No (harmonic) |
| $\displaystyle\sum \frac{1}{n^{1.001}}$ | $p = 1.001$ | ✓ Yes (barely!) |
| $\displaystyle\sum \frac{1}{n^{0.999}}$ | $p = 0.999$ | ✗ No |
Identifying P-Series
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^4}$ converges or diverges.
This is a p-series with $p = 4$.
Since $p = 4 > 1$:
$$\boxed{\text{The series converges}}$$
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ converges or diverges.
Rewrite: $\dfrac{1}{\sqrt{n}} = \dfrac{1}{n^{1/2}}$
This is a p-series with $p = \frac{1}{2}$.
Since $p = \frac{1}{2} < 1$:
$$\boxed{\text{The series diverges}}$$
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n^2}}$ converges or diverges.
Rewrite: $\dfrac{1}{\sqrt[3]{n^2}} = \dfrac{1}{n^{2/3}}$
This is a p-series with $p = \frac{2}{3}$.
Since $p = \frac{2}{3} < 1$:
$$\boxed{\text{The series diverges}}$$
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$ converges or diverges.
This is a p-series with $p = 1$.
Since $p = 1$ (not greater than 1):
$$\boxed{\text{The series diverges}}$$
This is the harmonic series — the famous boundary case.
P-Series with Coefficients
A constant multiple doesn't change convergence!
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} \frac{5}{n^3}$ converges or diverges.
$$\sum_{n=1}^{\infty} \frac{5}{n^3} = 5 \sum_{n=1}^{\infty} \frac{1}{n^3}$$
The inner sum is a p-series with $p = 3 > 1$, so it converges.
$$\boxed{\text{The series converges}}$$
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} \frac{-2}{n^{0.5}}$ converges or diverges.
$$\sum_{n=1}^{\infty} \frac{-2}{n^{0.5}} = -2 \sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$$
The inner sum is a p-series with $p = \frac{1}{2} < 1$, so it diverges.
$$\boxed{\text{The series diverges}}$$
(The negative sign doesn't save it!)
Recognizing Hidden P-Series
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} n^{-5}$ converges or diverges.
Rewrite: $n^{-5} = \dfrac{1}{n^5}$
This is a p-series with $p = 5 > 1$.
$$\boxed{\text{The series converges}}$$
Problem: Determine if $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ converges or diverges.
This is a p-series with $p = \frac{3}{2} = 1.5$.
Since $p = 1.5 > 1$:
$$\boxed{\text{The series converges}}$$
Problem: For what values of $k$ does $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{k+1}}$ converge?
This is a p-series with $p = k + 1$.
For convergence, we need $p > 1$:
$$k + 1 > 1$$ $$k > 0$$
$$\boxed{\text{Converges when } k > 0}$$
NOT a P-Series
Watch out for series that look similar but aren't p-series!
Problem: Is $\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n}$ a p-series?
No! In a p-series, $n$ is in the base: $\frac{1}{n^p}$
Here, $n$ is in the exponent: $\frac{1}{2^n}$
This is a geometric series with $r = \frac{1}{2}$, not a p-series.
$$\boxed{\text{Not a p-series — it's geometric (converges since } |r| < 1\text{)}}$$
Problem: Is $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$ a p-series?
No! A p-series has exactly $n^p$ in the denominator.
This has $n^2 + 1$, which is different.
$$\boxed{\text{Not a p-series — need comparison test or other methods}}$$
(It does converge, but you'd prove it by comparing to $\sum \frac{1}{n^2}$.)
Famous P-Series Values
Some p-series have known sums:
| Series | Sum |
|---|---|
| $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2}$ | $\dfrac{\pi^2}{6} \approx 1.645$ |
| $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^4}$ | $\dfrac{\pi^4}{90} \approx 1.082$ |
| $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^6}$ | $\dfrac{\pi^6}{945} \approx 1.017$ |
The pattern: $\displaystyle\sum \frac{1}{n^{2k}}$ always involves $\pi^{2k}$ (proved by Euler).
For odd powers like $\sum \frac{1}{n^3}$, the exact values are still unknown!
Common Mistakes and Misunderstandings
❌ Mistake: Confusing p-series with geometric series
Wrong: "$\sum \frac{1}{3^n}$ is a p-series with $p = 3$."
Why it's wrong: In $\frac{1}{3^n}$, the variable $n$ is in the exponent. In a p-series $\frac{1}{n^p}$, the variable $n$ is in the base.
Correct: $\sum \frac{1}{3^n}$ is geometric with $r = \frac{1}{3}$. A p-series example: $\sum \frac{1}{n^3}$.
❌ Mistake: Thinking $p = 1$ converges
Wrong: "$p = 1$ satisfies $p \geq 1$, so $\sum \frac{1}{n}$ converges."
Why it's wrong: The rule is $p > 1$, strictly greater. The harmonic series ($p = 1$) is the most famous divergent series!
Correct: $\sum \frac{1}{n}$ diverges because $p = 1$ is NOT greater than 1.
❌ Mistake: Applying p-series rule to non-p-series
Wrong: "$\sum \frac{1}{n^2 + n}$ is a p-series with $p = 2$, so it converges."
Why it's wrong: $n^2 + n \neq n^p$ for any $p$. The p-series rule only applies when the denominator is exactly $n^p$.
Correct: $\sum \frac{1}{n^2 + n}$ is NOT a p-series. You'd need partial fractions or comparison to determine convergence.
❌ Mistake: Forgetting to rewrite roots as exponents
Wrong: "$\sum \frac{1}{\sqrt{n}}$ — I don't see an exponent, so this isn't a p-series."
Why it's wrong: $\sqrt{n} = n^{1/2}$, so this IS a p-series with $p = \frac{1}{2}$.
Correct: Always convert roots to fractional exponents. $\frac{1}{\sqrt{n}} = \frac{1}{n^{1/2}}$, p-series with $p = 0.5 < 1$, diverges.
P-Series Convergence Test
The p-series test. Just check if p is greater than 1. Remember: p = 1 (harmonic series) diverges!
Variables:
- $p$:
- the exponent (a constant)
- $n$:
- the index variable
Common P-Series Reference
Key p-series to know: 1/n² converges to π²/6, 1/√n diverges (p=1/2), 1/n diverges (harmonic, p=1).
Variables:
- $p = 2$:
- converges
- $p = 1/2$:
- diverges
- $p = 1$:
- diverges (harmonic)
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