- Return 11.3 quiz
- Your homework for section 11.2 is due Tuesday, 10/13 (postponed).
- We'll still be working on 11.3 next time, so I'll also push back the 11.4 worksheet. No quiz on 11.4 next time.
- Continue working on your exam revisions! (Here's another link to Exam I). These are due next Thursday.
- Examples:
- #7, p. 554
- #8, p. 554 (compare to Taylor series error bound)
- Reflect on #36, p. 555. Recursive actions and definitions give rise to these types of geometric sums.
- Along the same lines: #45, p. 555
- Beautiful spiral "proofs" of some infinite series
- #33, p. 554
- #52, p. 556 (fractals!)
- concerns several different means of testing for convergence of
series with all positive terms (obviously, if we're dealing with series
of all negative terms, similar results are going to apply).
- We're not so concerned with the actual limits (that is the values
of the series) as we are with the convergence (or divergence) of the
series.
- Theorems:
The partial sums are terms in an increasing sequence, since the terms of the series are all positive.
Examples of the Integral Test:
- #9, p. 563
- #10, p. 563
Theorem 3 is just a corollary of Theorem 2, where the integrals are the obvious power functions:
- Comparison Test: #20, p. 563
This theorem says that, in the long run, one series has terms which are simply a constant times the terms in the other series. So, in the long term, one series is behaving like a scalar multiple of the other. And so if one converges, the other will, too.
Examples:
- Limit Comparison Test: #34, p. 563
- Divergence of the harmonic series (and dominoes!): #84, p. 565