Section 11.4

Series & Their Notations

This section pulls together everything from arithmetic and geometric sequences into the sum formulas you'll use directly on Exam 4. Infinite series problems — bouncing balls, repeating decimals — are popular exam questions, and the converge/diverge check is quick points once you know it.

⚡ Quick Summary

A series is what you get when you take a sequence (a list of numbers) and start adding them up. If your sequence is 2, 4, 6, 8, then the series is $2 + 4 + 6 + 8 = 20$ . A partial sum $S_n$ just means you stop adding after $n$ terms. Series come up whenever you need a running total — like adding up payments, scores, or distances.

Arithmetic series (partial sum)

S_n = \frac{n(a_1 + a_n)}{2}

Average of first and last term, times the number of terms.

Geometric series (partial sum)

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1

Multiply by $r$ , subtract, and almost everything cancels.

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The first question to ask with any series is: "Is this arithmetic (constant difference between terms) or geometric (constant ratio between terms)?" This tells you which formula to use. Arithmetic series use the "average of first and last, times the count" formula. Geometric series use a different formula involving the common ratio $r$ .

Geometric series have a superpower: if the ratio $r$ is between $-1$ and $1$ , you can add up infinitely many terms and still get a finite answer. The terms keep getting smaller and smaller, so the total settles on a specific number — it converges. That's how a bouncing ball can travel a finite total distance even though it bounces infinitely many times (in theory). If $|r| \geq 1$ , the terms don't shrink, so the sum keeps growing forever — it diverges.

On an exam, your first move is always: identify whether it's arithmetic or geometric. Then pick the matching formula. For infinite series, always check $|r|$ first — if it's not less than 1, just write "diverges" and move on. Don't overthink it!

Key Formulas

Arithmetic series (partial sum)

S_n = \frac{n(a_1 + a_n)}{2}

Average of first and last term, times the number of terms.

Geometric series (partial sum)

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1

Multiply by $r$ , subtract, and almost everything cancels.

Infinite geometric series

S_\infty = \frac{a_1}{1 - r}, \quad |r| < 1

Only converges when $|r| < 1$ ; diverges otherwise.

Sigma notation

\sum_{k=m}^{n} a_k = a_m + a_{m+1} + \cdots + a_n

Key Takeaways

✓Arithmetic series → common difference → use $S_n = \frac{n}{2}(a_1 + a_n)$
✓Geometric series → common ratio → use $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$
✓Infinite geometric series: converges only if $|r| < 1$ , then $S_\infty = \frac{a_1}{1-r}$
✓First step on any series problem: identify whether it's arithmetic or geometric

⚠️ Common Mistakes

✗Mixing up the arithmetic and geometric sum formulas — arithmetic uses $S_n = \frac{n}{2}(a_1 + a_n)$ ; geometric uses $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$
✗Jumping into a formula without first identifying whether the series is arithmetic (constant difference) or geometric (constant ratio)
✗Using the infinite series formula $S = \frac{a_1}{1 - r}$ without checking that $|r| < 1$ first — if $|r| \geq 1$ , just write "diverges"
✗In bouncing ball problems, counting the initial drop as a bounce — the first drop is just $h$ ; after that, each bounce goes up AND down (so you double those heights)

Next: Section 11.6 Builds on: 9.3

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9.3 Summation Notation

11.6 Binomial Theorem