Geometric series and the sum to infinity

What you are looking at

In a geometric sequence each term is the previous one multiplied by the common ratio, so $u_n = u_1 r^{n-1}$. Adding the first $n$ terms gives $S_n = \dfrac{u_1(1 - r^n)}{1 - r}$ for $r \neq 1$. The Bars & sums view draws each term as a green bar and joins the partial sums with a blue line, so you can see how much each new term adds. The Number line view shows the same sums as a chain of hops.

When $|r| < 1$, $r^n \to 0$ as $n$ grows, so $S_n$ approaches $S_\infty = \dfrac{u_1}{1 - r}$, drawn as a dashed yellow line. The part still missing after $n$ terms is $S_\infty - S_n = \dfrac{u_1 r^n}{1 - r}$, and the Table view lists it beside $u_n$ and $S_n$. When $|r| \geq 1$ the terms do not shrink, the partial sums have no finite limit and the visualiser reports that the series diverges.

Try this

1. 1Start from $u_1 = 8$, $r = 0.5$ and press Play n. Each new term covers half of the remaining distance to $S_\infty = 16$; after $10$ terms the gap is $\dfrac{16}{1024} = 0.015625$.
2. 2Change $r$ to $-0.5$. The partial sums now land alternately above and below $S_\infty = \dfrac{8}{1.5} = \dfrac{16}{3}$, closing in from both sides; the Number line view shows the hops changing direction.
3. 3Drag $r$ slowly towards $1$ and watch $S_\infty = \dfrac{u_1}{1 - r}$ grow while the partial sums take longer to get near it. At $r = 1$ every term equals $u_1$, so $S_n = n\,u_1$.
4. 4Hide the formulas and open the Table view. The $S_n$ column shows question marks, so you can predict each partial sum before revealing it.

Mistakes students make

• Using $S_\infty = \dfrac{u_1}{1 - r}$ when $|r| \geq 1$. The formula holds only for $|r| < 1$; otherwise there is no sum to infinity.
• Writing $u_n = u_1 r^n$. The first term is $u_1 r^0$, so the $n$th term is $u_1 r^{n-1}$.
• Mishandling a negative ratio. With $r = -0.5$, $1 - r = 1.5$, not $0.5$, so $S_\infty = \dfrac{u_1}{1.5}$.

Where it is taught

• IB Maths AA SL/HL 1.3 · Geometric sequences and series
• IB Maths AA SL/HL 1.8 · Infinite convergent geometric series
• Edexcel A-Level Maths (9MA0) · Pure Year 2 · Sequences and series: geometric series and sum to infinity
• AP Calculus BC Unit 10 · Working with geometric series (10.2)
• Greek Lykeio (Α΄ Λυκείου) · Algebra · Geometric progressions