We know what it means to add 2, 100, or even a lakh terms. What does it mean to add an infinite number of terms?

Step-by-step solution

Idea: Nobody can do infinitely many additions. Instead, add the first term, then the first two, then the first three, and so on, and watch where these ‘partial sums’ are heading. That target value is the sum of the infinite series.

1. Take Mādhava’s series π = 4 × (1 − 13 + 15 − 17 + …). Add terms one at a time: 4 × 1 = 4; 4 × (1 − 13) ≈ 2.667; then ≈ 3.467, ≈ 2.895, ≈ 3.340, ≈ 2.976, ≈ 3.284, ….1 mark
2. The running totals swing above and below 3.14159… and the swings get smaller and smaller. The value they get closer and closer to is the sum of the infinite series, here π. So adding an infinite number of terms means finding this limiting value.1 mark

Check: After 10 terms the total is about 3.042 and after 1000 terms about 3.1406: closer to π each time ✓.

Answer to write in the exam

Common mistakes that cost marks

• Thinking an infinite sum must be infinitely large. Here the terms shrink fast enough for the totals to settle on π.
• Stopping after a few terms and calling that ‘the sum’. A partial sum is only an approximation.
• Expecting the partial sums to rise steadily. In this series they go up and down, but the jumps shrink.

How this can come in the exam

Try one yourself

Write 0.333… as an infinite sum and say what value it approaches.

More questions like this

• It terminates: The division eventually leaves a remainder of 0. The decimal stops. 38 = 0.375. (Can you tell for which rational numbers the decimal will be terminating?)
• It repeats: The division never reaches a remainder of 0, but the sequence of digits begins to loop infinitely. 511 = 0.454545 … = 0.45.
• Try to find the decimal expansions of 103 and 1112. What do you observe about the repetition of the digits after the decimal point?
• Why do some rational numbers have repeating decimal representations? Imagine calculating 17 using long division. You are dividing by 7. What are the possible remainders at each step? They can only be 1, 2, 3, 4, 5, or 6 (why not 0?).
• The decimal expansion of pq will be terminating precisely when the prime factors of q are only 2, only 5 or both 2 and 5. Can you explain why?

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