Hop closer and closer

The idea

π is given exactly by Mādhava's infinite series π/4 = 1 – 1/3 + 1/5 – 1/7 + …, a limit approached by adding more and more terms.

Adding for ever

Mādhava of Sangamagrāma saw that π is not just a number to be approximated by fractions. It is a limit to be reached. He found the first exact formula for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + …

If we keep adding and subtracting these fractions for ever, what happens to the total?

• It grows bigger and bigger without end
• It gets closer and closer to one fixed value
• It jumps around with no pattern

Hop along the number line

Start at 0. The first hop is +1. Then the hops are −1/3, +1/5, −1/7, … Keep hopping and watch the total.

A limit to be reached

π is given exactly by Mādhava's infinite series π/4 = 1 − 1/3 + 1/5 − 1/7 + … Its value is a limit: the totals get closer and closer to it as more and more terms are added.

The totals land to the right of π/4, then to the left, and every hop is smaller. Using this series Mādhava found π to 11 decimal places: 3.14159265358.

Notes

π/4 = 1 − 1/3 + 1/5 − 1/7 + … is an exact formula for π: a limit approached by adding more and more terms.

Check yourself

The terms of the series are 1, 1/3, 1/5, 1/7, … What is the denominator of the 6th term?

Answer: 11

The denominators are 1, 3, 5, 7, 9, 11.

The totals so far: 1, then 1 − 1/3. What is the next term to add?

After 3 terms the total is about 0.867. Is this more or less than π/4, which is about 0.785?

What is the value of the infinite series 1 − 1/3 + 1/5 − 1/7 + …?

• + 1/5 — correct. Yes. The signs alternate (+, −, +, −, …) and the denominators go up by 2.
• − 1/5. After − 1/3 the sign turns back to +.
• + 1/4. The denominators are odd numbers: 1, 3, 5, 7, …
• Less. It has not reached π/4 yet. 0.867 is bigger than 0.785, so it is already past π/4.
• More. The total has gone past π/4 — correct. Yes. 1 − 1/3 + 1/5 makes 13/15, which is 0.867…, past π/4. The next hop, −1/7, brings it back below π/4.
• Exactly π/4. The total never equals π/4 after any finite number of terms. It only gets closer and closer.
• The total after the very last term. The series has no last term. It goes on for ever.
• It grows without end. The hops get smaller and smaller, so the total stays near π/4.
• The limit the totals get closer and closer to, which is π/4 — correct. Yes. We cannot finish adding, but we can see where the totals are heading.