Divergence test
If the terms of a series do not tend to 0, the series diverges. Terms tending to 0 is necessary, but does not guarantee convergence.
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Analogy
If you keep adding chunks that never get small, the pile must keep growing.
Example
Terms: 1/2, 2/3, 3/4, ...Limit of n/(n+1) is 1, not 0.Terms stay close to 1.So the series diverges.
In practice
A fast first check before any harder test.
Practice
Flashcards
- What does the divergence test say?If the terms do not tend to 0, the series diverges.
Questions people ask
If the terms go to zero, must the series converge?
No. The harmonic series 1 + 1/2 + 1/3 + and so on has terms tending to zero, yet it diverges. The test can only prove divergence. When the terms do tend to zero, you must try a different test to decide.
Why must the terms of a convergent series go to zero?
If the running total settles on a number, consecutive totals get very close together. Their difference is the next term, so that term must get tiny. Terms that stay near 1 would keep pushing the total up by about 1 every step.