1. Take absolute value: $\sum_{n=1}^{\infty} \frac{1}{n^2}$. 2. This is a convergent p-series (p=2). 3. Therefore, the series is absolutely convergent.

1. Take the absolute value: $\sum_{n=1}^{\infty} \frac{|\cos(n)|}{n^2}$. 2. Since $|\cos(n)| \leq 1$, compare to $\sum_{n=1}^{\infty} \frac{1}{n^2}$. 3. The p-series converges, so the original series is absolutely convergent.

1. Verify the series converges using Alternating Series Test. 2. Take the absolute value of the terms. 3. Show the absolute value series diverges. 4. Conclude it's conditionally convergent.

1. Alternating Series Test shows convergence. 2. Absolute value gives $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$, a divergent p-series (p=1/2). 3. Conditionally convergent.

1. Alternating Series Test shows convergence. 2. Absolute value gives $\sum_{n=1}^{\infty} \frac{n}{n^2 + 1}$, which diverges by Limit Comparison Test with $\frac{1}{n}$. 3. Conditionally convergent.

1. Take absolute value: $\sum_{n=1}^{\infty} \frac{|\sin(n)|}{n!}$. 2. Since $|\sin(n)| \leq 1$, compare to $\sum_{n=1}^{\infty} \frac{1}{n!}$. 3. Ratio Test shows $\sum_{n=1}^{\infty} \frac{1}{n!}$ converges. 4. Absolutely convergent.

1. Alternating Series Test shows convergence. 2. Absolute value gives $\sum_{n=2}^{\infty} \frac{1}{\ln(n)}$, which diverges by Comparison Test with $\frac{1}{n}$. 3. Conditionally convergent.

1. Take absolute value: $\sum_{n=1}^{\infty} \frac{n^2}{2^n}$. 2. Apply Ratio Test. 3. The series converges absolutely.

1. Alternating Series Test shows convergence. 2. Absolute value gives $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} + n}$, which converges by Limit Comparison Test with $\frac{1}{n^{3/2}}$. 3. Absolutely convergent.

1. Alternating Series Test fails since $\lim_{n \to \infty} \frac{n+1}{n} = 1 \neq 0$, so the series diverges. 2. No need to check absolute convergence.

A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges.

A series $\sum a_n$ is conditionally convergent if $\sum a_n$ converges but $\sum |a_n|$ diverges.

A series converges if the sequence of its partial sums approaches a finite limit.

A series diverges if the sequence of its partial sums does not approach a finite limit.

A harmonic series is a series of the form $\sum_{n=1}^{\infty} \frac{1}{n}$.

A series where the terms alternate in sign, often involving $(-1)^n$ or $(-1)^{n+1}$.

A p-series is a series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where p is a constant.

A test to determine convergence or divergence by comparing a given series to a known convergent or divergent series.

A test used to prove convergence of an alternating series if the absolute value of the terms decreases monotonically to zero.

An ordered list of numbers.

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 $\sum \frac{1}{n^p}$.