Essay for Mathematics

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The Mathematics for Essay 2
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
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