The Series Convergence Lab

Summation notation, arithmetic series, and finite/infinite geometric series

Walk half the distance to the wall. Then half of what is left. Then half of that. You take infinitely many steps and you never touch the wall. But the total distance you cover is finite, and you can compute it exactly. Adding up infinitely many things does not have to give you infinity.

Step 1 of 9

In Simulation 29, Company A paid $40,000 the first year with $2,000 raises each year for 30 years. A sequence lists terms. A series adds them up. What did you actually earn over the entire 30-year career?

The first year pays $40,000 and the last year pays $40,000 + 29(2,000) = $98,000. Add all 30 salaries together.

Total career earnings ($): (within $5,000)

Step 2 of 9

That was 30 numbers to add. Imagine doing that for 100 years, or 1,000. There has to be a shortcut. Look at these salary bars stacking up.

Step 3 of 9

Write the sum forwards, then write it backwards right below. Add each vertical pair.

Step 4 of 9

Now that you understand what a series is, here is the shorthand. Summation notation uses the Greek letter sigma to compress a sum into one expression.

$\displaystyle \sum_{k=1}^{30} [40{,}000 + (k-1)(2{,}000)]$

This says: "For k going from 1 to 30, add up 40,000 + (k - 1)(2,000)."

Translate this summation: what is the first term when k = 1?

Step 5 of 9

Now try a geometric series. Each term is multiplied by a constant ratio instead of adding a constant difference. Consider: $S = 3 + 6 + 12 + 24 + 48$. The ratio is $r = 2$.

The forwards-backwards trick from arithmetic series does not work here. Different structure needs a different trick.

Multiply the entire sum by $r = 2$:

Subtract the second line from the first. Watch what cancels.

Step 6 of 9

Now for the big question. Consider: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$ Each term is half the previous one. Add terms one at a time and watch the running total.

Step 7 of 9

The finite geometric sum formula is $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$. When $|r| < 1$ and $n$ gets huge, what happens to $r^n$?

Step 8 of 9

Sweep the common ratio and predict: does the geometric series converge (finite sum) or diverge (sum grows without bound)?

Common ratio $r$: 0.5

With $a_1 = 1$ and $r = $ 0.5, predict:

Step 9 of 9

Back to the bouncing ball from Simulation 29. A ball dropped from 10 feet rebounds to 75% of its height each bounce. The total distance traveled is a geometric series, and it converges.

The ball falls 10 feet, then bounces up 7.5 and falls 7.5, then bounces up 5.625 and falls 5.625, and so on. After the first drop, every bounce contributes distance twice (up and down).

$\text{Total} = 10 + 2(7.5) + 2(5.625) + 2(4.21875) + \ldots = 10 + 2 \cdot \frac{7.5}{1 - 0.75}$

What is the total vertical distance before the ball stops?

Explore Series

First term $a_1$: 1.0

Common ratio $r$: 0.50

Number of terms $n$: 10

Show summation notation

Partial sum: --

Stacking Bars

Running Total

Adjust the sliders to explore how the first term, common ratio, and number of terms affect geometric series behavior.

Try This

Repeating Decimals as Series

The repeating decimal $0.333\ldots = \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \ldots$

This is a geometric series. The first term is $a_1 = \frac{3}{10}$ and the common ratio is $r = \frac{1}{10}$.

Using the infinite sum formula $S = \frac{a_1}{1 - r}$, what does $0.333\ldots$ equal?

Drug Dosing: Steady State

A patient takes 200 mg of a medication every 24 hours. The body clears 70% of the drug each day, so 30% of each dose remains when the next dose arrives.

After the first dose: 200 mg. Right before the second dose: $200 \times 0.3 = 60$ mg remain, then 200 more are added, giving 260 mg. And so on.

The long-run steady-state amount right after a dose is a geometric series with $a_1 = 200$ and $r = 0.3$. What is the steady-state amount? (within 1 mg)

Steady-state amount (mg):

Bouncing Ball: Total Distance

A ball is dropped from 10 feet and rebounds to 80% of its height on each bounce. Find the total vertical distance traveled before the ball stops.

Careful: after the first drop, every bounce contributes distance twice -- once going up and once coming back down.

Total = first drop + 2 × (sum of all rebound heights)

The rebound heights form a geometric series: $8 + 6.4 + 5.12 + \ldots$ with $a_1 = 8$ and $r = 0.8$.

What is the total vertical distance? (within 0.5 feet)

Total distance (feet):

Distinguish between a sequence (a list of terms) and a series (a sum of terms)
Compute the sum of an arithmetic series using $S_n = \frac{n}{2}(a_1 + a_n)$
Read and write summation (sigma) notation
Derive and apply the geometric series formula $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$
Determine whether an infinite geometric series converges or diverges
Compute the sum of a convergent infinite geometric series using $S = \frac{a_1}{1 - r}$

Quick Check

An infinite geometric series has $a_1 = 12$ and $r = \frac{2}{3}$. What is the sum?

Instructor Notes

Teaching Notes

The Discover sequence builds from arithmetic series (Gauss's trick) through geometric series (multiply-and-subtract) to infinite geometric series (what happens as n grows). Each formula is derived, not given.
Step 6 is the key moment: students watch partial sums approach 2 and must commit to a prediction about whether the sum ever exceeds 2. This is where the concept of convergence becomes concrete.
Step 8 lets students discover the convergence condition $|r| < 1$ by testing multiple values of r, rather than being told the rule.
The connection between the finite formula and the infinite formula (step 7) is central: the infinite formula is just the finite formula with the vanishing term removed.

Common Student Errors

Believing that an infinite sum must be infinite. The stacking blocks visualization (step 6) directly addresses this.
Forgetting that the bouncing ball's first drop counts once while each rebound counts twice (up and down). The Challenge tier tests this specifically.
Confusing when to use the arithmetic vs. geometric formula. Arithmetic has a common difference; geometric has a common ratio.
Including $r = 1$ in the convergent range. At $r = 1$ every term equals $a_1$ and the sum grows without bound.
Misapplying the formula when $r = 1$ (division by zero).

Discussion Questions

Zeno's paradox says you can never reach the wall because you always have half the distance left. But you do reach the wall. Where does the paradox break down?
Why does a drug reach a steady state if the body clears a fixed percentage each day but not if it clears a fixed amount?
Can you think of other real-life processes that involve adding infinitely many things that get smaller?

Exam Connection

Standard exam items: given $a_1$ and $r$, compute the finite or infinite sum; determine convergence or divergence; convert a repeating decimal to a fraction.
Application problems involving total distance (bouncing ball) and drug accumulation are common.
Students should be able to write a series in summation notation and expand summation notation into a series.