Chapter 2 — Series Basics, Geometric Series, p-Series

By the end of this chapter you will:

1. Define series convergence rigorously via partial sums
2. Decide convergence of a geometric series and find its sum
3. Decide convergence of a p-series
4. Use the n-th term test as a one-look diverge-detector
5. Handle telescoping series

2.1 What is a Series? — The Partial-Sum Idea

Picture this: you are slicing an apple. First slice takes half ($\frac{1}{2}$), second takes half of what's left ($\frac{1}{4}$), third takes half again ($\frac{1}{8}$)…

Adding all the slices:
$$S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$

You sense the total approaches $1$ (because eventually the entire apple is sliced). That is what we mean by convergence of an infinite series.

Formal definition (the partial-sum approach):

$$\text{The series } \sum_{n=1}^{\infty} a_n \text{ converges} \iff \text{the partial sums } S_N = \sum_{n=1}^{N} a_n \text{ converge}$$

So:

1. Form the partial sum $S_N = a_1 + a_2 + \cdots + a_N$ (this is a sequence).
2. Check whether $\lim_{N\to\infty} S_N$ exists.

🔑 Key insight: a series is just the limit of a sequence of partial sums. Every tool from Chapter 1 carries over.

🎵 Audio 1 — Partial sums explained — audio/ch2_01_partial_sum.mp3

2.2 Geometric Series — A BC Must-Memorize

Form:
$$\sum_{n=0}^{\infty} a r^n = a + ar + ar^2 + ar^3 + \cdots$$

Here $a$ is the first term and $r$ is the common ratio.

Convergence test

Derivation (you should know this)

$$S_N = a + ar + ar^2 + \cdots + ar^{N-1}$$

Multiply both sides by $r$:
$$rS_N = ar + ar^2 + \cdots + ar^{N}$$

Subtract:
$$S_N - rS_N = a - ar^N \Rightarrow S_N = \frac{a(1 - r^N)}{1 - r}$$

When $\lvert r\rvert < 1$, $r^N \to 0$, so $S_N \to \dfrac{a}{1-r}$.

🎵 Audio 2 — Geometric series formula and derivation — audio/ch2_02_geometric.mp3

Examples — Recognize and Sum

Example 1: $\displaystyle\sum_{n=0}^{\infty} \frac{3}{2^n}$.

Recognize: $a = 3$, $r = \frac{1}{2}$. Since $\lvert r\rvert < 1$, converges. Sum $= \dfrac{3}{1 - 1/2} = 6$.

Example 2: $\displaystyle\sum_{n=1}^{\infty} \frac{2^n}{3^{n-1}}$.

Rewrite to align indices:
$$= \sum_{n=1}^{\infty} 2 \cdot \frac{2^{n-1}}{3^{n-1}} = \sum_{n=1}^{\infty} 2 \left(\frac{2}{3}\right)^{n-1}$$

First term ($n=1$) $= 2$, $r = \frac{2}{3}$. Sum $= \dfrac{2}{1 - 2/3} = 6$.

🔑 Strategy: when $r$ is hidden, factor out coefficients and shift indices to match $n=0$ or $n=1$.

⚠️ Trap: the starting index changes the first term $a$. The same general term $\frac{1}{2^n}$ has $a=1$ when starting from $n=0$ but $a=\frac{1}{2}$ when starting from $n=1$ — sums differ by a factor of 2.

2.3 p-Series — The Second Must-Memorize

Form:
$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$

Convergence test

Three classics (memorize)

🔑 Counterintuitive point: the harmonic series has terms going to 0, yet the sum diverges to infinity. So "terms tend to 0" does not guarantee convergence. The n-th term test only detects divergence — never convergence.

🎵 Audio 3 — p-series and the harmonic surprise — audio/ch2_03_pseries.mp3

Sketch of why

Compare the series with the integral $\int_1^\infty x^{-p}\,dx$. The integral converges precisely when $p > 1$, and the integral test (Chapter 3) makes this rigorous.

2.4 The n-th Term Test — One-Way Divergence Detector

Theorem: if $\lim_{n\to\infty} a_n \ne 0$ (including non-existent), then $\sum a_n$ diverges.

Intuition: to add infinitely many numbers and stop somewhere finite, the terms eventually have to shrink to 0. Otherwise you keep adding nonzero pieces, and the total flies off.

How to use it (notice the asymmetry!)

🔑 One-way theorem:

• $\lim a_n \ne 0 \Rightarrow$ series diverges ✅ (usable)
• $\lim a_n = 0 \Rightarrow$ series may or may not converge ❌ (no conclusion)

The harmonic series is the canonical reminder: terms $\to 0$ yet diverges.

Examples

Example 1: $\displaystyle\sum_{n=1}^{\infty} \frac{n}{2n+1}$.

Term limit $\to \frac{1}{2} \ne 0$, so diverges. ✅

Example 2: $\displaystyle\sum_{n=1}^{\infty} (-1)^n$.

$(-1)^n$ oscillates between $\pm 1$, no limit, so diverges. ✅

🎵 Audio 4 — One-way nature of the n-th term test — audio/ch2_04_nthterm.mp3

2.5 Telescoping Series

A telescoping series is one whose general term splits as $a_n = b_n - b_{n+1}$ (or some other adjacent-cancellation form).

Signature: when you write the partial sum, almost every term cancels with the next, leaving just a head and tail.

Example

Find $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$.

Step 1 — Partial fraction split:
$$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$$

Step 2 — Write the partial sum:
$$S_N = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{N} - \frac{1}{N+1}\right)$$

Almost everything cancels:
$$S_N = 1 - \frac{1}{N+1}$$

Step 3 — Take the limit:
$$\lim_{N\to\infty} S_N = 1 - 0 = 1$$

So the series converges to 1. ✅

🎵 Audio 5 — Telescoping pattern — audio/ch2_05_telescoping.mp3

🔑 Recognition cues: terms involving $\frac{1}{n(n+k)}$, $\ln\frac{n}{n+1}$, or $\sqrt{n+1}-\sqrt{n}$ — try splitting.

2.6 Mixed Examples (AP-Style)

Decide convergence; find the sum if it converges.

(a) $\displaystyle\sum_{n=0}^{\infty} \left(\frac{2}{5}\right)^n$

(b) $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$

(c) $\displaystyle\sum_{n=1}^{\infty} \frac{2n^2 + 1}{3n^2 - n}$

(d) $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+2)}$

Solutions:

(a) Geometric with $r = 2/5 < 1$, converges. Sum $= \dfrac{1}{1-2/5} = \dfrac{5}{3}$.

(b) p-series with $p = 3/2 > 1$, converges. (AP usually only asks for the verdict.)

(c) Term limit $= \dfrac{2}{3} \ne 0$, n-th term test → diverges.

(d) Telescoping: $\dfrac{1}{n(n+2)} = \dfrac{1}{2}\left(\dfrac{1}{n} - \dfrac{1}{n+2}\right)$.

Partial sum:
$$S_N = \frac{1}{2}\left[\left(1 - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \cdots\right]$$

Surviving terms: $\dfrac{1}{2}\left(1 + \dfrac{1}{2} - \dfrac{1}{N+1} - \dfrac{1}{N+2}\right) \to \dfrac{1}{2}\cdot\dfrac{3}{2} = \dfrac{3}{4}$. ✅

🎵 Audio 6 — Walkthrough of (a)–(d) — audio/ch2_06_examples.mp3

2.7 Trap Alerts ⚠️

1. The n-th term test only detects divergence — never convergence. When $a_n \to 0$ you must move on to other tests.
2. A geometric series's first term depends on the starting index. $\sum_{n=0}^\infty r^n = \frac{1}{1-r}$ vs. $\sum_{n=1}^\infty r^n = \frac{r}{1-r}$.
3. $\sum \frac{1}{n}$ diverges. This is one of the most-tested counterintuitive facts in BC.
4. Don't mis-cancel a telescoping sum. Always expand the first few terms and the last few terms before jumping to the limit.
5. $\lvert r\rvert = 1$ in a geometric series → diverges, regardless of sign.

2.8 Mnemonic

"Geometric → check $\lvert r\rvert$; p-series → compare with 1; term limit → divergence (one way); telescoping → split and cancel."

Media Inventory

🎵 Audio 7 — Chapter recap — audio/ch2_07_recap.mp3

End of chapter. Next: the full toolkit of convergence tests.