What is the general form of a p-series?
$\sum_{n=1}^{\infty} \frac{1}{n^{p}}$
What is the formula for the harmonic series?
$\sum_{n=1}^{\infty} \frac{1}{n}$
Define a p-series.
A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$, where p is a constant.
Define a harmonic series.
A p-series where p = 1, represented as $\sum_{n=1}^{\infty} \frac{1}{n}$.
What does it mean for a series to converge?
The sum of the infinite terms approaches a finite value.
What does it mean for a series to diverge?
The sum of the infinite terms does not approach a finite value; it tends to infinity or oscillates.
How do you determine convergence/divergence of $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$?
Identify p = 2. Since 2 > 1, the series converges.
How do you determine convergence/divergence of $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$?
Rewrite as $\sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$. Identify p = 1/2. Since 1/2 < 1, the series diverges.
How do you determine convergence/divergence of $\sum_{n=1}^{\infty} \frac{n^2}{n^3}$?
Simplify to $\sum_{n=1}^{\infty} \frac{1}{n}$. Identify p = 1. Since p=1, the series diverges (Harmonic Series).
Determine convergence/divergence of $\sum_{n=1}^{\infty} \frac{1}{n^{0.75}}$
Identify p = 0.75. Since 0.75 < 1, the series diverges.
Determine convergence/divergence of $\sum_{n=1}^{\infty} n^{-4}$
Rewrite as $\sum_{n=1}^{\infty} \frac{1}{n^4}$. Identify p = 4. Since 4 > 1, the series converges.
Determine convergence/divergence of $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^5}}$
Simplify to $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$. Identify p = 3/2. Since 3/2 > 1, the series converges.
Determine convergence/divergence of $\sum_{n=1}^{\infty} (\frac{1}{n})^3$
Rewrite as $\sum_{n=1}^{\infty} \frac{1}{n^3}$. Identify p = 3. Since 3 > 1, the series converges.
