Adding forever
Mādhava's infinite series for π
An infinite sum
In the 14th century, Mādhava of Sangamagrama found an exact formula for π. It does not use one fraction. It uses an infinite sum: π = 4 × (1 − 13 + 15 − 17 + …).
What happens to the total as you add more and more terms?
What this lesson covers
The idea
π = 4 × (1 – 1/3 + 1/5 – 1/7 + …); the value of such an infinite sum is the value approached ever more closely as more and more terms are added.
An infinite sum
In the 14th century, Mādhava of Sangamagrama found an exact formula for π. It does not use one fraction. It uses an infinite sum: π = 4 × (1 − 1/3 + 1/5 − 1/7 + …).
What happens to the total as you add more and more terms?
- It grows bigger and bigger without end
- It gets closer and closer to a fixed value
- It jumps around with no pattern
Add and watch
Add terms to Mādhava's sum and watch the total.
Closer and closer
π = 4 × (1 − 1/3 + 1/5 − 1/7 + …). The value of such an infinite sum is the value the totals get closer and closer to as more and more terms are added.
To express an irrational number you cannot use a single fraction: you need an infinite sum. You will learn more about infinite series in higher grades.
Notes
π = 4 × (1 − 1/3 + 1/5 − 1/7 + …). The value of an infinite sum is the value approached more and more closely as more terms are added.
Check yourself
What does it mean to add an infinite number of terms?
The terms are 1, 1/3, 1/5, 1/7, … What is the denominator of the fifth term?
Answer: 9
The denominators go up by 2: 1, 3, 5, 7, 9.
The series is 1 − 1/3 + 1/5 − 1/7 + … What comes next?
Why can no single fraction give π exactly?
- Add until you get tired and stop. We look for the value that the totals approach, not where we stop.
- Find the value the totals get closer and closer to as more terms are added — correct. Yes. We cannot finish adding, but the totals approach a value.
- The total becomes infinite. Here the total stays close to π. It does not grow without end.
- − 1/9. After − 1/7 the sign changes back to +.
- + 1/8. The denominators are 1, 3, 5, 7, 9, …: odd numbers.
- + 1/9 — correct. Yes. The signs alternate and the denominators go up by 2.
- Because π is irrational — correct. Yes. That is why Mādhava used an infinite sum.
- Because π is a very big number. π is about 3.14, which is small. The reason is that π is irrational.
- Because fractions cannot have 3 in them. Fractions can have any whole numbers in them. The reason is that π is irrational.