For series $\sum a_n$ and $\sum b_n$ with $0 \le a_n \le b_n$, if $\sum b_n$ converges, then $\sum a_n$ converges. If $\sum a_n$ diverges, then $\sum b_n$ diverges.

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

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

The geometric series $\sum_{n=0}^{\infty} ar^n$ converges if $|r| < 1$ and diverges if $|r| \ge 1$.

Find a series $\sum b_n$ whose convergence/divergence is known, and show that $0 \le a_n \le b_n$ (for convergence) or $a_n \ge b_n$ (for divergence).

Find a series $\sum b_n$ and compute $\lim_{n\to\infty} \frac{a_n}{b_n}$. If the limit is finite and positive, the series behave alike.

It requires finding a suitable inequality, which can be difficult. It's inconclusive if the inequality goes the wrong way.

The limit must be finite and positive. If the limit is 0 or infinity, the test is inconclusive.

It ensures that the inequalities used in the theorems are valid. If terms are negative, the comparison may not hold.

L'Hopital's Rule can be used to evaluate the limit $\lim_{n\to\infty} \frac{a_n}{b_n}$ when it results in an indeterminate form.

To determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known.

To ensure that the comparison is valid. If terms are negative, the inequalities used in the tests may not hold.

If a larger series converges, a smaller series must also converge. If a smaller series diverges, a larger series must also diverge.

If two series have similar end behavior, they will either both converge or both diverge.

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

If $|r| < 1$, the geometric series converges. If $|r| \ge 1$, the geometric series diverges.

Choose a series that has similar terms and known convergence/divergence, often a p-series or geometric series.

When it is easy to establish a direct inequality between the terms of the two series.

When it is difficult to establish a direct inequality, but the limit of the ratio of the terms is easy to compute.

That $b_n$ grows much faster than $a_n$.

Direct Comparison: Requires direct inequality ($a_n \le b_n$). Limit Comparison: Compares the limit of the ratio of terms. Direct Comparison: More straightforward when applicable. Limit Comparison: Useful when direct inequality is hard to establish.

p-series: Form $\sum \frac{1}{n^p}$, converges if $p > 1$. Geometric series: Form $\sum ar^n$, converges if $|r| < 1$. Both: Useful for comparison tests. p-series: Depends on exponent p. Geometric series: Depends on common ratio r.

$\sum b_n$ converges: Implies $\sum a_n$ converges. $\sum a_n$ diverges: Implies $\sum b_n$ diverges. The first shows convergence, the second shows divergence.

Finite positive number: Both series behave alike (both converge or both diverge). Zero or infinity: The test is inconclusive and a different comparison series must be found.

$p>1$: $\sum b_n$ converges, useful for showing convergence of a smaller series. $p \le 1$: $\sum b_n$ diverges, useful for showing divergence of a larger series.

Direct Comparison: Compares series to another series. Integral Test: Relates series convergence to integral convergence. Both: Used to determine convergence/divergence. Integral Test: Requires function to be continuous, positive, and decreasing.

r=0.5: Converges because |0.5| < 1. r=2: Diverges because |2| > 1. Convergence depends on the absolute value of r being less than 1.

If $a_n > b_n$ and $\sum a_n$ converges, the test is inconclusive. If $a_n > b_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Comparison tests are typically used for series with positive terms. Alternating series often require the Alternating Series Test.

Obvious inequality: Application is straightforward. Requires manipulation: More complex, requires algebraic skills to establish the inequality.