PROBLEM SET XII
“I KNOW YOU’RE SAD TO SEE THE PROBLEM SETS END …”
DUE FRIDAY,  DECEMBER
… so I’ve made this one extra hard!
Just kidding. Rather, you’re going to see all the ways in which your hard work over the
course of the semester pays off. With just a little more effort, you’re going to prove all
kinds of cool stuff now.
Let’s start with a few power series:
∑1
exp(z) :=
zk ;
k!
k≥0
cos(z) :=
∑ (−1)k
k≥0
sin(z) :=
(2k)!
z2k ;
∑ (−1)k
z2k+1 ;
(2k + 1)!
k≥0
cosh(z) :=
∑
k≥0
sinh(z) :=
∑
k≥0
atan(z) :=
1 2k
z ;
(2k)!
1
z2k+1 ;
(2k + 1)!
∑ (−1)k z2k+1
;
2k + 1
∑ 1 (2k)
c(z) :=
zk ;
k+1 k
k≥0
k≥0
h(z) :=
∑ bk
k≥0
k!
zk .
(Here, bk denotes the Bernoulli numbers from Exercise .)
Exercise . Compute the radii of convergence of each of these power series.
Exercise . Show that the power series labeled atan(z) converges to arctan(z) (as defined in Problem Set VIII) for real z inside the radius of convergence. Show that the series
atan(1) converges as well and that in fact you already have a name for the limit:
π ∑ (−1)k
=
.
4
2k + 1
k≥0


DUE FRIDAY,  DECEMBER
(Here π is as in Problem Set VIII.) is is called the Gregory-Leibniz series for computing π, but it really shouldn’t be. (ere’s a rule of thumb in mathematics that an object
is neer named for the person who invented it, and this fits with that.) is series was
discovered by Madhava of Sangamagrama in the th century, a brilliant mathematician,
who, appallingly, many people have not heard of !
Exercise . Prove that for any x ∈ R, one has the usual properties of cos and sin:
| cos(x)| ≤ 1 and | sin(x)| ≤ 1,
and
cos(x + 2π) = cos(x) and
sin(x + 2π) = sin(x).
(Hint: how do you get π involved?) Prove any other properties of cos and sin that you
like using these power series.
Exercise . Prove that cosh and sinh as defined here agree with the definitions I gave in
class:
1
1
cosh(x) := (exp(z) + exp(−z)) and sinh(z) := (exp(z) − exp(−z)),
2
2
and show also that for any x ∈ R,
cosh(ix) = cos(x) and
sinh(ix) = i sin(x).
Exercise . Use the previous exercise to show that for any real number θ,
exp(iθ) = cos(θ) + i sin(θ).
Use this to construct a C∞ bijection (0, +∞) × [0, 2π) .
call polar coordinates.
C − {0}. is is what they
Exercise . Show that cos and cosh, as functions R . R, are both of C-type (Exercise
). ose polynomials pn from Problem Set VI are called Chebyshev polynomials.
Exercise . Prove the following identity:
n−1
∏
sin(kπ/n) = 21−n n.
k=1
Exercise⋆ . Use the power series for exp to show that
√
∫ +∞
∫ b
π
2
2
.
exp(−t ) dt := lim
exp(−t ) dt =
2
b→+∞ 0
0
Exercise . Show that within the radius of convergence, the power series c can be written
√
1 − 1 − 4z
.
c(z) =
2z
Exercise . Show that within the radius of convergence, the power series h(z) agrees
with the function from Exercise .
PROBLEM SET XII
“I KNOW YOU’RE SAD TO SEE THE PROBLEM SETS END …”

Exercise . Take a minute to reflect on how far we’ve come this semester. ink about
what you could do at the beginning of the semester and compare it with what you can
do now. I hope that all of you are experiencing a real sense of empowerment. e great
news is: if you’re willing to work at it, that will continue for the rest of your life. You’ll
spend your whole life looking back at wonder at all the things you didn’t know just a few
months ago!
ank you all for a fun semester! I really enjoyed doing math with you this fall. Please have
a loely winter break, and I hope you’ll join me in the spring for more fun. Even if you decide
not to take ., though, please don’t hesitate to ask me for any help/advice you might need!
S   
I know, this probably isn’t the time of year when I can expect you to solve problems this
hard. at’s OK. But these are really fun, and if you get a chance over the winter break,
please give them some thought!! If you do solve some of them, let we know!
Exercise⋆⋆ . Prove that Euler’s constant e = exp(1) is transcendental (Exercise ) in
the following manner. Suppose
a0 + a1 e + · · · + am em = a0 exp(0) + a1 exp(1) + · · · + am exp(m) = 0
for some integer coefficients ai with a0 ̸= 0. For any prime number p, write
xp−1 (x − 1)p (x − 2)p · · · (x − m)p
,
(p − 1)!
g(x) :=
and set
∑
mp+p−1
G(x) :=
g(k) (x).
k=0
Check the following:
|g(x)| <
mmp+p−1
;
(p − 1)!
d
(exp(−x)G(x)) = − exp(−x)g(x);
dx
∫ j
aj
exp(−x)g(x) dx = aj G(0) − aj exp(−j)G(j).
0
Deduce that
m
∑
j=0
∫
aj exp(j)
j
exp(−x)g(x) dx = −
0
m mp+p−1
∑
∑
j=0
aj g(i) (j).
i=0
Now show that g(i) (j) is always an integer, and show that it is divisible by p unless j ̸= 0
and i ̸= p − 1. If p > m, show that g(p−1) (0) is not divisible by p. Finally, show that if p
is sufficiently large, then
m mp+p−1
∑
∑
−
aj g(i) (j)
j=0
i=0

DUE FRIDAY,  DECEMBER
is nonzero, yet


∑
∫ j
m
m
m+2 )p−1
∑
≤
 exp(m) (m
a
exp(j)
exp(−x)g(x)
dx
|a
|
< 1,
j
j
(p − 1)!
0
j=0
j=0
a contradiction.
Exercise⋆⋆ . Make sense of and prove the following identity:
+∞
∑
π
(−1)n
=
.
sin(πz) n=−∞ z − n
Use this to show that
)
∞ (
∏
z2
sin(πz) = πz
1− 2 .
n
n=1
Show that the infinite product G(s) from Exercise  makes sense and converges for s ∈
C − {0, −1, −2, . . . }, and use the above to show that
G(s)G(1 − s) =
πs2
.
sin(πz)
e function Γ(s) = G(s)/s is called the Gamma function.
Exercise⋆ . Find a power series
∑ ak
z2k
(2k)!
k≥1
with radius of convergence π/2 such that for any x ∈ R with |z| < π/2, one has
∑
ak xk ,
tan(x) =
k≥1
where tan : (−π/2, π/2) . R is the inverse of arctan (Exercise ). (Hint: the coefficients ak involve the Bernoulli numbers; the same is true for all the trig functions except
sin and cos. Use the recurrence from Exercise  to find them in the power series expansion of tan.)
Exercise⋆⋆ . Show that for any n ≥ 1, the series
∑ 1
ζ(2n) =
k2n
k≥1
converges to
b2n (2π)2n
.
2(2n)!
(Hint: use polar coordinates to prove the following identity for any z ∈ C − {0} such
that |z| < 1/2:
∑ 1
π
1
2z
=
− ,
2
2
z −k
tan(zπ) z
(−1)n+1
k≥1
PROBLEM SET XII
“I KNOW YOU’RE SAD TO SEE THE PROBLEM SETS END …”

and use the power series you found for tan.) e function ζ(s) is called the Riemann zeta
function. is identity, originally due to Euler, is only the beginning. Describing ζ as interesting is like saying Antarctica is chilly.