Comparison test
If , then converges by comparison: dominating a series by a convergent non-negative series forces convergence.
Proposition 1.28 (Comparison Test)
If for all sufficiently large , then
Proof
Let and be the partial sums of and . Because , both sequences of partial sums are increasing. Since , we have for all , so
is increasing and bounded above by , hence converges by the monotone convergence theorem. Only the tail matters, so the hypothesis may hold merely for all sufficiently large .
Example
converges:
where converges with partial sums . Comparison with this convergent dominating series settles the question.
Related
Stated in
- Proposition 1.28 (Comparison Test)§1.4 Series and Convergence Tests