**Power series: everyones favorite functions are
polynomials!**

Unpacking the concepts:

(this stuff might look familiar, but it's good to go through).

1. For what x does

converge? converge conditionally? converge
absolutely?

What kind of series is this?

2. For those x for whichconverges,
what does it converge to?

3. For each of the following x's,

(i) estimateby calculating the first few
partial sums.

(ii) what is?

(iii) how far out in the sequence of partial sums do you have to go to get
within 1=100

of the actual value?

Conjecture: the sequence of partial sums ofconverge
faster when____ .

4. Turn to page 799 in your book, and read up on testing for convergence. For
what x

to the following series converge absolutely? converge conditionally? diverge?

5. I have a secret function in a box, about which I claim
two things: (1) it can me written

as an infinite power series, and (2) when I take it's derivative, I get the
function back

(a) What is this super secret series? (Hint: all power series look like

Take this thing's derivative and set it equal to the
series and solve for the coef-

ficients . Look for patterns.)

(b) For what x does this series converge?

(c) What's another function you know which satisfies part (2)?

(d) For those x which causes this series to converge, what does it converge to?

6. e is** irrational**. Recall that a rational number is one which can be
written as the

ratio of integers, and that an irrational number is one that can't. In general,
proving

that a number is rational is a lot easier than proving that it isn't| the former
just

requires you to find the appropriate fraction, whereas the latter can't be done
by

example (you can't try all possibilities, since there are infinitely many!!).

Use the series expansion of e^{x} to prove that 1/e is not rational, and therefore
neither

is e.

(hint:

(a) Assume you can write e as p/q, where p and q are integers, and are reduced

(there are no factors that p and q have in common. So what's 1/e?

(b) Plug in x = -1 into the series expansion for e^{x}, and use some partial sum

to estimate 1/e. What kind of series do you get? What's the error of your

estimate? What does that mean about the distance between q/p and s_{n}?
Write

this inequality down.

(c) You can pick an n large enough so that there is something you can multiply
both

sides by to get integers on both sides. How large does n have to be? How do you

clear all the fractions in a useful way? What can you conclude?

(d) Find a contradiction of your assumption in part (a). What does this mean? )

7. The other day, as Ruth and Zajj were cleaning up, an alien appeared in the
WES

classroom. The alien told them \I'm thinking of two integers, both bigger than 3
and

smaller than 97. I will tell their sum to Ruth and their product to Zajj." The
alien

did this and then disappeared. The following conversation occurred.

Ruth: You dont know what the numbers are.

Zajj: Well, I didn't, but now I do.

Ruth: Now I do too!

Assuming Zajj and Ruth weren't using their psychic powers, determine what the
two

integers were.