The purpose of these notes are to explain some of the mathematics behind Essay 2. Your own essay should not just repeat these arguments but have a more geometric ﬂavor. Write about how you can physically place the blocks. You may assume basic facts about geometric sums and series. Let r be any real number and let n be a non-negative integer. The sum 1 + r + r2 + · · · + rn

(1)

is a geometric sum and the inﬁnite series

1 + r + r2 + · · · + rn + · · ·

(2)

is a geometric series.

Suppose further that r = 1. Then the geometric sum (1) can be computed by the formula

1 − rn+1

.

1 + r + r2 + · · · + rn =

1−r

This fact, which you may assume, is easily proved proved by mathematical induction. Now suppose that |r| < 1. Then limn−→∞ rn = 0 which means the geometric series (2) 1

by the preceding equation. We write

converges to

1−r

1 + r + r2 + · · · + rn + · · · =

1

1−r

(3)

to indicate that the series converges and to designate the limit of the sequence of partial sums.

Your essay will involve the geometric series

1+

1

11

+ + ··· + n + ···.

24

2

(4)

1

11

1

Since | | < 1, it follows by (3) that (4) converges and 1 + + + · · · + n + · · · = 2. The 2

24

2

1

1

1

,

,

, . . . Your essay involves anaDeluxe blocks are cubes with side lengths 1, 2

3

5

lyzing the sum of their side lengths

1+

11111111

1

+ + + + + + + + ··· +

+ ···.

23456789

16

The preceding series is called the harmonic series. Think of the terms of the geometric series (4) as markers for grouping terms of the harmonic series as follows: 1+

1

11

1111

1

1

+ ( + ) + ( + + + ) + ( + ··· + ) + ···.

2

34

5678

9

16

(5)

We will ﬁnd an overestimate and an underestimate for the sum of the terms in each of the parenthesized groups. You will see a pattern emerging in our calculations: 1=

11

11

11

1

+>+>+=,

22

34

44

2

1111

1111

1111

1

+++>+++>+++=,

4444

5678

8888

2

1

1

1

1

1

1

1

1

1

>

+ ··· +

= 8( ) = ,

1 = + · · · + = 8( ) > + · · · +

8

8

8

9

16

16

16

16

2

.

.

.

1=

Using (5) and our underestimates, we see that

111111111

1

+ + + + + + + + + ··· +

+ ···

234567889

16

1

11

1111

1

1

= 1 + + ( + ) + ( + + + ) + ( + ··· + ) + ···

2

34

5678

9

16

1111

> 1 + + + + + ···.

2222

1+

Thus the partial sums of the harmonic series grow without bound which is expressed by 1+

1111

+ + + + · · · = ∞.

2345

Below is a formal proof of the the fact that the sums of terms in parenthesized groupings 1

lie between and 1. You should not include the proof in your essay; the mathematics of 2

your essay is to be treated informally. Observe that the terms of a parenthesized group are 1

1

given by n

, . . . , n+1 for some n ≥ 1.

2 +1

2

Lemma 1 Let n be a positive integer. Then

1

1

1