- Last time we wrapped up Sect. 11.6: Absolute convergence test (pdf, pdf summary), including the two new tests: ratio and root.
All these tests are "self-referential" -- limits of the ratio or roots
of terms of the series may tell us whether the series converges or not.
We're going to be making serious use of these tests as we move
forward. I hope that you're feeling okay about those!
- The next section we consider, Sect. 11.8: Power series (pdf, pdf summary)
takes us in a different, and very important, direction:
What if we replace the constants in the series with functions?
- It's easy enough to think of an example like this: what if I
change the size of the room, and then run Zeno's experiment? I can take
an $x/2$ step, then $x/4$, then $x/8$, etc. How far will I have gone if
I add up the entire infinite series?
\[
\sum_{k=1}^\infty\frac{x}{2^k} = x\sum_{k=1}^\infty\frac{1}{2^k} = x
\]
No big deal, right?
- Now what if we take steps that are more complicated functions of
$x$? E.g. what if we take steps that are polynomial functions of the
step number, e.g.
\[
\sum_{k=1}^\infty x^k
\]
Well actually, if we let $x=\frac{1}{2}$, this is Zeno's problem:
\[
\sum_{k=1}^\infty \left(\frac{1}{2}\right)^k
=
\sum_{k=1}^\infty \frac{1}{2^k}
\]