98 CHAPTER 5. INFINITE SERIES OF NUMBERS

12. You can define infinite series of complex numbers in exactly the same way as infiniteseries of real numbers. That is w = ∑

∞k=1 zk means: For every ε > 0 there exists N

such that if n ≥ N, then |w−∑nk=1 zk| < ε. Here the absolute value is the one which

applies to complex numbers. That is, |a+ ib| =√

a2 +b2. Show that if {an} is adecreasing sequence of nonnegative numbers with the property that limn→∞ an = 0and if ω is any complex number which is not equal to 1 but which satisfies |ω|= 1,then ∑

∞n=1 ωnan must converge. Note a sequence of complex numbers, {an + ibn}

converges to a+ ib if and only if an → a and bn → b. See Problem 6 on Page 64.There are quite a few things in this problem you should think about.

13. Suppose limk→∞ sk = s. Show it follows limn→∞1n ∑

nk=1 sk = s.

14. Using Problem 13 show that if ∑∞j=1

a jj converges, then it follows

limn→∞1n ∑

nj=1 a j = 0.

15. Show that if {pi}∞

i=1 are the prime numbers, then ∑∞i=1

1pi= ∞. That is, there are

enough primes that the sum of their reciprocals diverges. Hint: Let π (n) denote thenumber of primes less than equal to n,

{p1, ..., pπ(n)

}. Then explain why

n

∑k=1

1k≤

(n

∑k=1

1pk

1

)· · ·

(n

∑k=1

1pk

π(n)

)≤

π(n)

∏k=1

11− 1

pk

≤π(n)

∏k=1

e2/pk = e2∑π(n)k=1

1pk

and consequently why limn→∞ π (n) = ∞ and ∑∞i=1

1pi= ∞.

16. Verify the allegation about the Euclidean norm |x| ≡(

∑pk=1 |xk|2

)1/2that Fp with

the Euclidean norm yields the same Cauchy sequences, compact sets, and open andclosed sets as Fp with the norm ∥·∥.

17. Suppose S is an uncountable set and suppose f (s) is a positive number for each s∈ S.Also let Ŝ denote a finite subset of S. Show that

sup

{∑s∈Ŝ

f (s) : Ŝ⊆ S

}= ∞