Welcome back to Professor Baker's Math Class! In this session, we transition from looking at sequences (lists of numbers) to Infinite Series (the sum of those numbers). This is a pivotal moment in Calculus II, as determining whether a sum adds up to a finite number (converges) or grows without bound (diverges) is central to the rest of the course.

1. What is a Series?

A series is denoted by the summation notation $\sum_{n=1}^{\infty} a_n$. To understand if an infinite sum makes sense, we look at its partial sums, denoted as $s_n$. If the sequence of partial sums tends toward a specific number $s$ (i.e., $\lim_{n \to \infty} s_n = s$), we say the series is convergent and its sum is $s$. Otherwise, it is divergent.

One classic example covered in the notes is the Telescoping Series, such as $\sum \frac{1}{n(n+1)}$. By using partial fractions, we can show that intermediate terms cancel out, leaving us with a clean finite sum.

2. The Geometric Series

A major focus of this lecture is the Geometric Series, which takes the form:

$$\sum_{n=1}^{\infty} ar^{n-1} = a + ar + ar^2 + \dots$$

The behavior of this series depends entirely on the common ratio, $r$.

  • If $|r| < 1$, the series converges. The sum is given by the formula: $$S = \frac{a}{1-r}$$.
  • If $|r| \ge 1$, the series diverges.

In the class notes, we looked at several examples, including one where $a=5$ and $r=-2/3$. Since $|-2/3| < 1$, that series converged to 3. However, be careful with algebraic manipulation! Another example involved powers of 2 and 3 that simplified to a ratio of $4/3$. Since $4/3 > 1$, that series diverged.

3. The Test for Divergence

This is one of the most important logic checks in series. The Test for Divergence states:

If $\lim_{n \to \infty} a_n$ does not exist or if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ is divergent.

Warning: The reverse is not true! If the limit is zero, the series might converge or it might diverge. You must use other tests to find out.

4. The Integral Test and p-Series

Finally, we introduced the connection between series and improper integrals. If a function $f(x)$ is continuous, positive, and decreasing, then the series $\sum a_n$ behaves the same way as the integral $\int_{1}^{\infty} f(x) \, dx$.

This leads us to the p-series test, which is a shortcut you will use frequently:

$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$
  • Converges if $p > 1$.
  • Diverges if $p \le 1$.

This explains why the Harmonic Series (where $p=1$) diverges, despite the terms getting smaller and smaller.

Keep practicing identifying which test to use for which problem. Mastering these rules now will make the upcoming chapters much smoother. You've got this!