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3 Responses
Quick eyeball rule: if each term is a fixed multiple r of the previous one (that’s a geometric series), then it’s easy-if |r| < 1 it sums to a number, if |r| ≥ 1 it blows up. So 1 + 1/2 + 1/4 + … has r = 1/2 and converges; neat trick: call the sum S, then 1/2 S = 1/2 + 1/4 + …, subtract to get S − 1/2 S = 1, so S = 2. Meanwhile 1 + 2 + 4 + … has r = 2, so it just keeps exploding. More generally, if the terms don’t go to zero, forget it-no sum. If they do go to zero, you’re usually fine; for example a series like 1 + 1/2 + 1/3 + … still converges, just painfully slowly. Alternating signs with shrinking terms also tends to settle down (like 1 − 1/2 + 1/4 − 1/8 + …, which lands at 2/3). When in doubt, compare consecutive terms: ratio less than 1 means tame, greater than 1 means runaway.
Love this! The quick look-at-it rule: if it’s a geometric series, check the common ratio r between terms. If |r| < 1, it actually adds up to a finite number (sum = first term divided by 1 − r); if |r| ≥ 1, it doesn’t converge. So 1 + 1/2 + 1/4 + … has r = 1/2 and sums to 2, while 1 + 2 + 4 + 8 + … has r = 2 and shoots off to infinity. A good first gate for any series: do the terms go to 0? If not, there’s no way the sum can settle down. If it’s not obviously geometric, a handy “grown-up version” of the ratio idea is the ratio test: look at the limit of |a_(n+1)/a_n|; if it’s less than 1, you converge; greater than 1, you diverge; equal to 1, try another test. The way I finally remembered this was thinking about a bouncing ball: if each bounce is a fixed fraction of the last, it eventually comes to rest; if each bounce is the same or bigger, it goes forever-same vibe as r < 1 versus r ≥ 1. I once mixed up these two series on a quiz until I scribbled “terms must → 0!” at the top of my paper like a lighthouse. For a clear walk-through, this Khan Academy intro to geometric series is great: https://www.khanacademy.org/math/algebra/sequences/alg-geometric-series/a/geometric-series-introduction.
The quick “look-at-it” rule I use is: check what the terms themselves are doing. If the terms get smaller and smaller and actually go to 0, then the series will settle down to a finite sum. If the terms don’t go to 0 (or they’re growing), then there’s no way the total can level off-it’ll blow up. I like to think of it like pouring smaller and smaller spoonfuls of water into a cup: if the spoonfuls shrink to nothing, you’ll eventually fill the cup to some level; if the spoonfuls don’t shrink to nothing, you’ll keep overfilling.
For your two examples: in 1 + 2 + 4 + 8 + … the terms are getting bigger, not smaller, so there’s no chance the sum converges. In 1 + 1/2 + 1/4 + 1/8 + … the terms are halving each time, so they’re marching straight down to 0, which means the series does have a sum. A quick worked example: call S = 1 + 1/2 + 1/4 + 1/8 + …. Then 2S = 2 + 1 + 1/2 + 1/4 + …, and subtracting gives 2S − S = (2 + 1 + 1/2 + …) − (1 + 1/2 + 1/4 + …) = 2, so S = 2. That little subtraction trick makes it feel very concrete.