Sum the infinite series 9 + 3 + 1 + …

Question

Solve in your head. What is 9 + 3 + 1 + 	frac13 + 	frac19 + \dots (forever)? Aim: under 10 seconds.

Accepted Answer

Each term is one-third of the one before, so this is an infinite geometric series with first term a = 9 and ratio r = 	frac13. The infinite sum is 	ext{sum} = \frac{a}{1 - r} = \frac{9}{1 - 	frac13} = \frac{9}{2/3} = 9 	imes \frac{3}{2} = 13.5. Intuition: dividing by 1 - r asks "how many first-terms is the whole tail worth?" With r = 	frac13 the answer is 	frac{1}{1 - 1/3} = 	frac32 of the first term, so 9 	imes 	frac32 = 13.5. The famous cousin is r = 	frac12, where the sum is exactly 2a: that is why 1 + 	frac12 + 	frac14 + \dots = 2. Takeaway. An infinite geometric series with r…

Sum the infinite series 9 + 3 + 1 + …

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