Adding endlessly many terms can still give a finite total, provided the terms shrink fast enough.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The condition
An infinite geometric series converges exactly when the common ratio satisfies |r| < 1. Otherwise the terms never fade and the sum runs away.
The sum
When it converges, S = a₁ ÷ (1 − r).
Where the formula comes from
In the finite sum a₁(1 − rⁿ) ÷ (1 − r), the term rⁿ shrinks toward zero when |r| < 1, leaving a₁ ÷ (1 − r).
A picture
Walk half the remaining distance to a wall, again and again. The steps total 1 wall-length without ever passing it.
Repeating decimals
Every repeating decimal is such a series. 0.333... is 0.3 + 0.03 + 0.003 + ..., which sums to 1/3.
When adding forever gives a finite answer
An infinite geometric series converges when |r| < 1, and its sum is a₁/(1 − r). The terms shrink fast enough that the total settles rather than growing without bound.
Why the condition on r
If |r| ≥ 1 the terms do not shrink towards zero, so the partial sums keep growing and no limit exists. Checking |r| before applying the formula is not optional — the formula is meaningless outside that range.
It resolves an ancient puzzle
Zeno argued that crossing a room requires infinitely many steps and so is impossible. The steps form a geometric series with r = 1/2 summing to a finite distance. Infinitely many things can add to something finite.
Repeating decimals are geometric series
0.333… is 3/10 + 3/100 + 3/1000 + …, a geometric series with r = 1/10, summing to 1/3. This is why every repeating decimal is rational, proved rather than asserted.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Diego Add 8 + 4 + 2 + 1 + ...
Diego wants the total of this endless list.
- Step 1
Each term is half the one before, so the common ratio is 0.5.
Step 4: Your Turn
Practice makes it stick.
The Total
Problem 1 of 2
First term 6, ratio 0.5. What is the infinite sum?
The Bouncing Ball
Problem 2 of 2
A ball is dropped from 10 m and each bounce reaches 0.5 of the previous height. What is the total of all the bounce heights after the drop, in metres?
m
Forever, Finitely
1 of 8
First term 4, ratio 0.5. Infinite sum?
2 of 8
First term 9, ratio 0.5. Infinite sum?
3 of 8
First term 1, ratio 0.8. Infinite sum?
4 of 8
First term 3, ratio 0.25. Infinite sum?
5 of 8
Ratio 2. Does the series converge? Enter 1 for yes or 0 for no.
6 of 8
0.5 + 0.05 + 0.005 + ... What is the sum, as a decimal to three places?
7 of 8
Which ratios give a convergent infinite series?
8 of 8
First term 20, ratio 0.5. Infinite sum?
Step 5: Quick Check
Show what you know.
Question 1 of 2
First term 10, ratio 0.5. What is the infinite sum?
Question 2 of 2
When does an infinite geometric series converge?
What You Learned
- An infinite geometric series converges exactly when |r| < 1.
- Its sum is a₁ ÷ (1 − r).
- Every repeating decimal is one of these series in disguise.