Advanced High-School Mathematics

346 CHAPTER 6 Inferential Statistics

thatE must be infinite.

(b) Here we’ll give a nuts and bolts direct approach.^10 Note first

that the expectationE is given by

E=

∑∞

k=0

(2k+ 1)C(k)

22 k+1

, whereC(k) =

1

k+ 1

Ñ

2 k

k

é

, k= 0, 1 , 2 ,....

(i) Show thatC(k) =

2 · 6 · 10 ···(4k−2)

(k+ 1)!

, k≥1 (This is a

simple induction).^11

(ii) Conclude thatC(k) =

1

k+ 1

∏k

m=1

(

4 −

2

m

)

.

(iii) Conclude thatC(k)2−(2k−1)=

2

k+ 1

∏k

m=1

(

1 −

1

2 m

)

.

(iv) Using the fact that lnx > x−1, show that ln

(

1 − 21 m

)

>

−m^1 , m= 1, 2 ,...

(v) Conclude that

∏k

l=1

(

1 −

1

2 l

)

= e

ln

∏k

l=1(

1 − 21 l)

= e

∑k

l=1

ln( 1 − 21 l)

> e

−

∑k

l=1

(^1) l

e−(1+lnn)=

1

ne

(see Exercise 5 on page 269)

(vi) Finish the proof thatE=∞by showing that the series

for E is asymptotically a multiple of the divergent har-

monic series.

22. Here are two more simple problems where Catalan numbers ap-
    pear.

(^10) I am indebted to my lifelong friend and colleague Robert Burckel for fleshing out most of the
details. 11
This still makes sense ifk= 0 for then the numerator is the “empty product,” and hence is 1.