P-series

[the attached proof generalizes
 the proof that \sum 1/n = infty]
 compare to the integral test, using log

that is:
- show that the harmonic series diverges, using integral
  - generalize to integral test,
    and apply to p-series
- show that the harmonic series diverges, using attached
  - generalize
    and apply to p-series

-start with a concrete EG-!

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note:
both the integral proof and the $2^n$ proof also compute
-how quickly- the partial sums grow.
It's really analysis: we don't just -abstractly- show something
happens, but we -estimate- -how quickly- it happens.
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the key result is:
define an ``concave down increasing series'' to be:
where $a_n$ are positive and decreasing
(an increasing series is iff $a_n$ are positive;
 that is, iff the partial sums are increasing)
[can put ``semi'' everywhere here]
- a concave down increasing series $\sum a_k$
  converges iff the series
  $\sum 2^k a_{2^k}$
  converges.
Proof:
  Both these series are increasing, so they converge iff they're bounded.
  $$\sum_{k=0}^{2^n} a_k \geq \sum_{k=0}^n 2^k a_{2^k}$$
    [just group the terms: it's (1/3 + 1/4) >= (1/4 + 1/4)]
  so if the original series converges, the $2^n$ one does too.
  $$\sum_{k=0}^{2^n} a_k \leq 2\sum_{k=0}^n 2^k a_{2^k}$$
    (again, group the other way)
  [he uses the notation $t_k \geq s_n$ and $t_k \leq 2s_n$]
  So they're simultaneously bounded or unbounded
    ($t_k \to \infty$ iff $s_n$ does).

Incidentally, we showed a bit more: in the limit,
  $2\sum a_k \sum 2^k a_{2^k} \geq \sum a_k$
  so they actually converge to about the same number (within 2 of each
  other)
[easy exercise: give examples when this is sharp;
  obviously all examples are of these forms.]

Now apply this to p-series and get comparison to geometric.
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actually, the basic facts about series converging are pretty boring;
hence, they should be -proven- by the students,
not -shown- to them.
EG:
- "convergence => terms go to zero"
- semi-increasing sequence converges <=> bounded
- etc.


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make sure to distinguish between a -series- and its -image-
exercise: give two different series that have the same image
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A series is -exactly- a function
Show it: the series 1, -1, 1, -1, etc.
is exactly the -function
  n    0  1  2  3  4  5  6 ...
$a_n$  1 -1  1 -1  1 -1  1 ...
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p-series are a great EG of series we can prove converge, easily,
but -can't- compute what they converge to!
(well, we can for even naturals, but that's -work-)

honestly, geometric series are one of the few ones you can compute.
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log:
first EG of a homomorphism (well, so is $e^x$; actually an -iso-)
1: show properties from the definition in terms of integral
  (multiplication, identity, inverse, division)
  (change of variables/u-substitution is -key- here)
2: show that the properties (identity, inverse, division)
   follow -formally- from multiplication

This is how homomorphisms should be introduced:
do it in the case of log (to show that it -is natural- (no pun
intended, I swear)), -then- generalize
  (also, log is good b/c it's -work- to show that it's a homo!)
