Calculus: Analytic Geometry and Calculus, with Vectors

(^630) Series
Let e > 0. Choose an integer N such that Isk - s4 < e/2 when n > N
N
let C = I Isk - s4. Then, when 0 < r < 1,
k=0

(10) Jf (r) - sl < (1 - r)C + (1 - r) I I- sl rkSi

k=N+1

< (1 - r)C + (1 - r) rk

<(1-r)C+

k=N+1

and

If we choose S such that (1 - r)C < e/2 when 1 - S < r < 1, we will have

(11) If(r)-sI<e (1-S<r<1).

This gives (3) and the theorem is proved.

24 Tell why the Abel theorem of the preceding problem implies that

m

(x22+3242+...1=1_22+3242+

and hence that part (g) of Problem 2 implies that

('1 log (t i t)

dt = 1 - 22 + 32 -^42 + ...

0

25 Some of the most honorable parts of mathematics involve connections

between the Riemann zeta function and prime numbers. Deriving a basic

formula is a good exercise for us. Let pi, P2, p8, denote in order the prime

numbers 2, 3, 5, 7, 11, 13,.. Euclid proved that the set of primes is infinite.

We use the fundamental theorem of arithmetic which says that if is is an integer

for which n > I and if pk is the greatest prime factor of is, then is is uniquely

representable in the form

n = pla,p2a:... pka,,

where the exponents X1, a2, , Xk are nonnegative integers. For example,

504 = 21325171. Show that, when k is a positive integer and s > 1,

(1)

1 - 2

(2)^1 +3,"+9,+27,+

(3)

1 I

(^1) - 5

(4) i 1 + + + +

1 - - pk Pk Pa'

pk