In which we study another notion of convergence of a sequence, and see how it relates to our existing definition.
- Definition of a Cauchy sequence.
- Lemma 4: Let be a convergent sequence of real numbers. Then it is a Cauchy sequence. We saw that this followed easily from the definitions.
- Lemma 5: Let be a Cauchy sequence of real numbers. Then the sequence is convergent. We first proved that the sequence is bounded, then applied the Bolzano-Weierstrass theorem to obtain a convergent subsequence, then showed that in fact the whole sequence must be convergent.
- Definition of convergence of a series.
- Lemma 6: Let , , , … and , , , … be complex numbers.
- If and both converge, then so does for any complex numbers and .
- ‘Initial terms do not affect convergence.’ That is, if there is some such that for , then either and both converge or they both diverge.
To prove (i), we simply used the definition (being careful to work with partial sums). We’ll prove (ii) next time; I encourage you to try to prove it yourself before then.
- I handed out the first examples sheet.
Understanding today’s lecture
You could pick some examples of sequences to check directly whether they converge and whether they are Cauchy sequences. Make sure that you find ‘non-examples’ as well as examples! Is the sequence a Cauchy sequence? What happens if you try to extend the ideas from today’s lecture to the complex numbers?
Here’s another relevant Tricki article, this time about proving that a sequence converges by finding a convergent subsequence. Inevitably Wikipedia has something to say on the subject of Cauchy sequences.
Preparation for Lecture 4
- I give you two series and , and I tell you that converges, and that for all . Must the series converge?
- If I give you a series , and I tell you that converges, does that necessarily mean that converges?
- If I give you a series such that converges, does that necessarily mean that converges?