First, take the absolute value of the terms in the series, i.e., consider $\sum |a_n|$.

If $\sum |a_n|$ converges, then $\sum a_n$ is absolutely convergent.

If $\sum |a_n|$ diverges, but $\sum a_n$ converges, then $\sum a_n$ is conditionally convergent.

It's often easier to determine absolute convergence first. If a series is absolutely convergent, you don't need to check for conditional convergence.

The Alternating Series Test can be used to show that an alternating series converges. If the absolute value of that series diverges, then the original series is conditionally convergent.

Comparison tests, like the Direct Comparison Test, can be used to determine if $\sum |a_n|$ converges or diverges, thus helping to establish absolute convergence.

Since $-1 \leq \sin(n) \leq 1$, taking the absolute value means $|sin(n)| \leq 1$. This is useful for comparison tests.

A harmonic series is $\sum_{n=1}^{\infty} \frac{1}{n}$, and it's a classic example of a divergent series. It's often used for comparison.

Absolutely convergent series have better error estimation properties than conditionally convergent series, as rearranging terms in a conditionally convergent series can change its sum.

Take the absolute value of the terms and then apply convergence tests. If the absolute value converges, the series is absolutely convergent.

If $a_n$ is decreasing and $\lim_{n \to \infty} a_n = 0$, then $\sum (-1)^n a_n$ converges.

If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $a_n \geq b_n \geq 0$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.

Let $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. If $L < 1$, the series converges absolutely. If $L > 1$, the series diverges. If $L = 1$, the test is inconclusive.

Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. If $L < 1$, the series converges absolutely. If $L > 1$, the series diverges. If $L = 1$, the test is inconclusive.

If $\sum |a_n|$ converges, then $\sum a_n$ converges.

If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.

If $f(x)$ is continuous, positive, and decreasing on $[1, \infty)$, then $\sum_{n=1}^{\infty} f(n)$ and $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.

If a series is absolutely convergent, then any rearrangement of the series converges to the same sum.

Absolute: $\sum |a_n|$ converges. Conditional: $\sum a_n$ converges, but $\sum |a_n|$ diverges.

$\sum \frac{1}{n}$: Diverges (Harmonic). $\sum \frac{(-1)^n}{n}$: Conditionally Converges (Alternating Harmonic).

$\sum \frac{1}{n^2}$: Converges (p-series, p=2). $\sum \frac{(-1)^n}{n^2}$: Absolutely Converges.

Absolute: Ratio, Root, Comparison. Conditional: Alternating Series Test, then check absolute value for divergence.

Absolutely: Rearranging doesn't change the sum. Conditional: Rearranging can change the sum.

Direct Comparison: Directly compare terms. Limit Comparison: Compare the limit of the ratio of terms.

For $p > 1$, the series converges. For $p \le 1$, the series diverges.

Absolute convergence: Series converges even with absolute values. Divergence: Series does not approach a finite limit.

$\frac{1}{n}$ approaches 0 slower than $\frac{1}{n!}$. $\sum \frac{1}{n}$ diverges, while $\sum \frac{1}{n!}$ converges.

Alternating Series Test: Tests convergence of alternating series. p-series test: Tests convergence of series of the form $\sum \frac{1}{n^p}$.