10/7 Direct comparsion test

Definition: The limit comparison test states that if the limit of the ratio of two series converges to a finite, non-zero number, then both series have the same behavior (both converge or both diverge).
Example of the Limit Comparison Test:
- Consider the series $\sum \frac{1}{n}$ and the series $\sum \frac{2}{n}$.
- The limit of their ratio is given by:
  $\lim<em>{n \to \infty} \frac{\frac{1}{n}}{\frac{2}{n}} = \lim</em>{n \to \infty} \frac{1}{2} = \frac{1}{2}$
- Since the limit converges to a finite and non-zero number, both series diverge as their behavior is similar.
If one series converges or diverges, the other does as well, leading to the conclusion that they behave similarly. This follows from understanding that if the ratios of terms of two series tend to a constant, their sums will tend to behave similarly.

Definition: The direct comparison test states that if $an \leq bn$ for all terms in the series and $\sum bn$ converges, then $\sum an$ also converges. Conversely, if $\sum an$ diverges, then $\sum bn$ also diverges. This test is only applicable for positive series.
Example of the Direct Comparison Test:
- Let $a_n = \frac{1}{3n^2 + 4n + 5}$.
- Compare with $b_n = \frac{1}{n^2}$, which is a convergent p-series (with p = 2).
- We can show:
- For $\sum a_n$, since $3n^2$ dominates as $n \to \infty$, we have:
  $3n^2 + 4n + 5 \geq 3n^2$
- Therefore:
  $a_n \leq \frac{1}{3n^2}$
- Since $\sum \frac{1}{3n^2}$ converges, it follows from the direct comparison test that $\sum a_n$ also converges.

Limit Comparison Test: Suitable for comparing two series where limits are involved. Easier but requires limits to be computed, which could be tricky.
Direct Comparison Test: Directly compares series using inequalities. It simplifies calculations as it avoids limits, but requires more clever manipulations.

Pros of Limit Comparison Test:
- Always applicable as long as series are positive.
- Less algebra compared to direct comparison when properly set up.
Cons of Limit Comparison Test:
- May require tedious limit computations.
- Not all series lend themselves to clean limits.
Pros of Direct Comparison Test:
- Avoids complex limit calculations, making it easier to apply directly.
- Generally faster with clever observations leading to valid comparisons.
Cons of Direct Comparison Test:
- Requires meticulous attention to ensure correct series relationships.
- Sometimes results in inconclusive results if proper fractions or limits aren't chosen.

The approach to both tests requires establishing the asymptotic behavior of the sequences involved. Intuition plays an important role here: if you can identify dominant terms in series, you can reason about their convergence/divergence behavior more effectively.
For rational sequences $\frac{f(n)}{g(n)}$, observe how alterations to numerator or denominator affect their sizes:
1. Increasing the numerator increases the value of the fraction.
2. Decreasing the numerator decreases the value of the fraction.
3. Increasing the denominator decreases the value of the fraction.
4. Decreasing the denominator increases the value of the fraction.

Example 1: Series $\sum_{n=1}^{\infty} \frac{1}{3n^2 + 4n + 5}$:
- Using direct comparison with $b_n = \frac{1}{n^2}$ leads to a conclusion of convergence.
Example 2: Series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{3n - 2}}$:
- Intuitively behaves like $\frac{1}{\sqrt{n}}$, which diverges, supporting that the original series diverges. Validate with direct comparison or by limit comparison.
Example 3: Series $\sum_{n=1}^{\infty} \frac{n^2 + 2n - 1}{4n^5 - n + 2}$:
- The dominating terms yield behavior similar to $\frac{1}{n^3}$, allowing us to use either test for convergence.