Essay for Mathematics
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
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