One destination, three journeys
All three sequences tend to zero, but their errors decrease at different rates. The destination alone does not tell us the speed.
A visual math lesson
The same limit. Different speeds.
Three sequences. One limit. Very different journeys.
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What this lesson covers
- Compare three sequences that all approach zero.
- Use a logarithmic view to reveal their rates of decay.
- Separate a fixed tolerance from a threshold that shrinks with n.
The core takeaway
A fixed tolerance asks whether a sequence converges. A shrinking threshold can also demand a particular speed.
The lesson, in words
An equivalent explanation of the essential visual reasoning. Follow along, or read at your own pace.
Watch the distance
The curves compare 1/√n, 1/n, and 1/n². For large n, 1/√n stays farther from zero, while 1/n² gets close much faster.
1/√n → 0; 1/n → 0; 1/n² → 0
Change the scale
A logarithmic view separates curves that seem to flatten together on an ordinary axis. For a power law n to the power −p, the slope on log–log axes is −p. A steeper negative slope means faster decay.
Keep the target fixed
Imagine a fixed ring around the limit a. Convergence means that, for every positive ring radius ε, there is an index N after which every sequence point stays inside that ring. N can depend on ε.
For every ε > 0, there is N such that n ≥ N implies |x(n) − a| < ε.
Now shrink the target
A ring of radius 1/n shrinks as the sequence advances. Eventually staying inside this ring guarantees convergence, but demands more than convergence alone. The sequence 1/√n converges to zero and still stays outside the ring for n > 1. The sequence 1/n² eventually lies inside it. Purple and blue identify compared sequences; the red ring is the threshold.
1/n² < 1/n < 1/√n for n > 1
Compare the two criteria
Requiring eventual entry into every fixed ring of radius 1/k, for every positive integer k, is equivalent to convergence. Requiring eventual entry into the moving ring of radius 1/n is sufficient for convergence but is not necessary.
Remember the distinction
For the two criteria in this lesson, ask whether the ring radius is fixed while n advances, or shrinks with n. That difference separates a test for convergence from a stronger test that also constrains speed.
A closer look
