Day 38 in MAT229

This weekend we have our third exam, which will cover material through alternating series.

• Worksheet #7:
  1. key (.nb)
  2. key (.pdf)

• Lab #9:
  1. key (.nb)
  2. annotated key
  3. video overview

What's this all about?

Suppose I have the series \[ \sum_{k=1}^\infty\frac{1}{2^k} = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots \] The terms represent "room distances", and I start from the far wall.

Suppose I have a sandwich, and want to hand it to someone outside in the hallway, and that I have to be able to reach out to give it to them (they can't stick their hand in).

Then I have to get close enough for my arm to go through the door frame. How many terms will I need of the series to hand over the sandwich?

1. My arm length is the tolerance;
2. $n$ is the number of terms $a_k$ I'll need to go to get that close;
3. and $S_n$ will be the distance I've traveled at the moment I hand over the sandwich.

So $|S-S_n|<{\textrm{arm's length}}$

means taking $n$ terms and getting to within range to hand over the sandwich.

• Absolutely convergent: A series is called absolutely convergent if the series of absolute values $\sum |a_n|$ converges.

• Conditionally convergent: A series is called conditionally convergent if the series is convergent but not absolutely convergent.

• If the series is absolutely convergent, you can re-arrange terms willy-nilly: e.g. the series \[ \sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{k+1}\right) = \sum_{k=1}^\infty\frac{1}{k(k+1)} \]

  Note that if a positive-termed sequence is convergent, then it is absolutely convergent.

  So we can split this absolutely convergent series into even and odd terms, for example: \[ \sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{k+1}\right) = \sum_{k=1,odd}\left(\frac{1}{k}-\frac{1}{k+1}\right) + \sum_{k=1,even}\left(\frac{1}{k}-\frac{1}{k+1}\right) \] But notice that we cannot write \[ \sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{k+1}\right) = \sum_{k=1}^\infty\frac{1}{k} - \sum_{k=1}^\infty\frac{1}{k+1} \] We cannot break up the terms $a_k$ themselves and then re-arrange those (both series on the right are divergent -- harmonic).

• If the series is only conditionally convergent, you cannot re-order the terms: e.g. \[ \sum_{k=1}^\infty(-1)^{k+1}\frac{1}{k} \] We cannot write the conditionally convergent series as \[ \sum_{k=1}^\infty(-1)^{k+1}\frac{1}{k} = \sum_{k=1,even}^\infty\frac{1}{k} - \sum_{k=1,odd}^\infty\frac{1}{k} \] Both the series on the right are divergent.

\[ .467467... = \frac{4}{10}+\frac{6}{100}+\frac{7}{1000}+ \frac{4}{10000}+\frac{6}{100000}+\frac{7}{1000000}+ \ldots \] We group terms in threes to produce \[ .467467... = \frac{467}{1000}+\frac{467}{1000^2}+ \ldots \] from which we obtain \[ .467467... = \frac{467}{1000}\sum_{k=0}^{\infty}\left(\frac{1}{1000}\right)^k = \frac{467}{1000}\frac{1}{1-\frac{1}{1000}} = \frac{467}{999} \] Alternatively, we can think of \[ .467467... = .400400... + .0600600... + .007007... \] (completely re-ordering the terms; but this is an absolutely convergent series, and so) \[ .467467... = \frac{400}{1000}\sum_{k=0}^{\infty}\left(\frac{1}{1000}\right)^k + \frac{60}{1000}\sum_{k=0}^{\infty}\left(\frac{1}{1000}\right)^k + \frac{7}{1000}\sum_{k=0}^{\infty}\left(\frac{1}{1000}\right)^k \] or \[ .467467... = \frac{400}{999} + \frac{60}{999} + \frac{7}{999} = \frac{467}{999} \]

1. Ratio test: Let $\{a_k\}$ be a sequence and assume that the following limit exists: \[ r=\lim_{k\to \infty}\left|\frac{a_{k+1}}{a_k}\right| \]
   1. if $r<1$, then the series $\sum a_k$ converges absolutely.
   2. if $r>1$, then the series $\sum a_k$ diverges.
   3. if $r=1$, the test is inconclusive.
   This result says that eventually the ratio of successive terms is effectively constant, r: the terms of the sequence approach a "common ratio" as $k\to\infty$. What kind of series looks like that? A geometric series:

   \[ \sum_{k=0}^{\infty}{c}r^k \]

   The ratio test is effectively a self-referrential comparison test: we compare terms of $a_n$ with other terms $a_{n+1}$ (rather than with some other series).

2. Root test: Let $\{a_k\}$ be a sequence and assume that the following limit exists: \[ r=\lim_{k\to \infty}\sqrt[k]{|a_k|} \]
   1. if $r<1$, then the series $\sum a_k$ converges absolutely.
   2. if $r>1$, then the series $\sum a_k$ diverges.
   3. if $r=1$, the test is inconclusive.

   This result says that eventually the absolute values of the terms are effectively equal to $|a_k|={r^k}$: what kind of series looks like that? A geometric series!

   Notice that, once again, limits of sequences plays an important role. Series are just sums of sequences, after all; we're focused on how terms behave (root test), how successive terms behave (ratio test), or how partial sums behave.