Welcome to Professor Baker's Calc 2 Class!
In this session, we delved into sections 11.4 through 11.6, exploring various techniques to determine the convergence or divergence of infinite series. Let's review the key concepts and examples discussed.
Direct Comparison Test
The Direct Comparison Test helps us determine the convergence or divergence of a series by comparing it to another series whose behavior is already known. Suppose we have two series, $\sum a_n$ and $\sum b_n$, with positive terms.
- If $\sum b_n$ converges and $a_n \leq b_n$ for all $n$, then $\sum a_n$ also converges.
- If $\sum b_n$ diverges and $a_n \geq b_n$ for all $n$, then $\sum a_n$ also diverges.
Example: Consider the series $\sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3}$. We can compare this to the $p$-series $\sum_{n=1}^{\infty} \frac{5}{2n^2}$. Since $2n^2 + 4n + 3 > 2n^2$, we have $\frac{5}{2n^2 + 4n + 3} < \frac{5}{2n^2}$. The series $\sum_{n=1}^{\infty} \frac{5}{2n^2} = \frac{5}{2} \sum_{n=1}^{\infty} \frac{1}{n^2}$ is a $p$-series with $p = 2 > 1$, so it converges. Therefore, by the Direct Comparison Test, $\sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3}$ also converges.
Limit Comparison Test
The Limit Comparison Test is another powerful tool. Suppose $\sum a_n$ and $\sum b_n$ are series with positive terms. If $$ \lim_{n \to \infty} \frac{a_n}{b_n} = c $$ where $c$ is a finite number and $c > 0$, then either both series converge or both diverge.
Example: Let's analyze $\sum_{n=1}^{\infty} \frac{2}{\sqrt{n} + 2}$. We can compare it with $\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}}$. Then, $$ \lim_{n \to \infty} \frac{\frac{2}{\sqrt{n} + 2}}{\frac{2}{\sqrt{n}}} = \lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n} + 2} = \lim_{n \to \infty} \frac{1}{1 + \frac{2}{\sqrt{n}}} = 1 $$ Since $\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}} = 2 \sum_{n=1}^{\infty} \frac{1}{n^{1/2}}$ is a $p$-series with $p = \frac{1}{2} \leq 1$, it diverges. Thus, by the Limit Comparison Test, $\sum_{n=1}^{\infty} \frac{2}{\sqrt{n} + 2}$ also diverges.
Alternating Series Test
The Alternating Series Test applies to series of the form $\sum_{n=1}^{\infty} (-1)^{n-1} b_n$ where $b_n > 0$. If the following conditions are met, the series converges:
- $b_{n+1} \leq b_n$ for all $n$ (i.e., the terms are decreasing).
- $\lim_{n \to \infty} b_n = 0$.
Example: Consider the alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$. Here, $b_n = \frac{1}{n}$. We have $b_{n+1} = \frac{1}{n+1} \leq \frac{1}{n} = b_n$, and $\lim_{n \to \infty} \frac{1}{n} = 0$. Thus, by the Alternating Series Test, the alternating harmonic series converges.
Absolute and Conditional 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.
Example: The series $\sum_{n=1}^{\infty} \frac{\cos n}{n^2}$ is absolutely convergent because $\sum_{n=1}^{\infty} \left| \frac{\cos n}{n^2} \right| \leq \sum_{n=1}^{\infty} \frac{1}{n^2}$, and the latter converges (it's a $p$-series with $p=2 > 1$).
Ratio Test
The Ratio Test is useful for series involving factorials or exponentials. Given a series $\sum a_n$, let $$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$
- If $L < 1$, the series is absolutely convergent.
- If $L > 1$, the series is divergent.
- If $L = 1$, the test is inconclusive.
Root Test
The Root Test is another test for convergence/divergence. Given a series $\sum a_n$, let $$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$$
- If $L < 1$, the series is absolutely convergent.
- If $L > 1$, the series is divergent.
- If $L = 1$, the test is inconclusive.
Keep practicing and you'll master these convergence tests in no time! Good luck!