Practice question · Multiple choice
The infinite geometric series 1 + 1/2 + 1/4 + 1/8 + … has infinitely many positive terms and sums to exactly 2. Why does adding infinitely many positive numbers not necessarily give an infinite total?
Hints
- Compute the first four partial sums. What are they approaching?
- Ask what "the sum of infinitely many terms" is defined to mean.
Show the answer
C. Because the sum is the limit of partial sums, here approaching 2.
Why
An infinite sum is a definition rather than an infinitely long computation: the limit of 1, 1.5, 1.75, 1.875, each halving the remaining distance and never overshooting. So 2 is exact. What decides it is how fast terms shrink, and going to zero is not enough, the harmonic series does that and diverges. It is also the resolution of Zeno's paradox.
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