Welcome back!
July 5
7.1–7.6. Sigma notation, supremum, infimum, and the
definition of the integral

Warm-up: sums
Compute
1.
4X
i=2
(2i + 1)
2.
4X
i=2
2i + 1
3.
4X
j=2
(2i + 1)
Write these sums with ⌃ notation
1. 15 + 25 + 35 + 45 + . . .+ 1005
2.
2
42
+
2
52
+
2
62
+
2
72
+ . . .+
2
N2
3. cos 0 cos 1 + cos 2 cos 3 + . . .± cos(N + 1)
4.
1
0!
+
1
2!
+
1
4!
+
1
6!
+ . . .+
1
(2N)!
5.
1
1!
1
3!
+
1
5!
1
7!
+ . . .+
1
81!
6.
2x3
4!
+
3x4
5!
+
4x5
6!
+ . . .+
999x1000
1001!
Re-writing sums
1.
100X
i=1
tan i
50X
i=1
tan i =
X
2.
NX
i=1
(2i 1)5 =
N1X
i=0
3.
"
NX
k=1
xk
#
+
"
NX
k=0
k xk+1
#
=
264 X
k=
xk
375 +
Hint: Write out the sums on the left hand side first, simplify if
possible, then write them back into sigma notation.
Telescopic sum
Calculate the exact value of
137X
i=1

1
i
1
i + 1

Hint: Write down the first few terms.
Calculate the exact value of
10,000X
i=1
1
i(i + 1)
Telescopic sum
Calculate the exact value of
137X
i=1

1
i
1
i + 1

Hint: Write down the first few terms.
Calculate the exact value of
10,000X
i=1
1
i(i + 1)
Warm up: suprema and infima
Find the supremum, infimum, maximum, and minimum of the
following sets (if they exist):
1. A = [1, 5)
2. B = (1, 6] [ (8, 9)
3. C = {2, 3, 4}
4. D =
n1
n
: n 2 Z, n > 0
o
5. E =
⇢
(1)n
n
: n 2 Z, n > 0

6. F = {2n : n 2 Z}
Suprema from a graph
Calculate, for the function g on the interval [0.5, 1.5]:
1. supremum 2. infimum 3. maximum 4. minimum
Empty set
1. Does ; have an upper bound ?
2. Does ; have a supremum?
3. Does ; have a maximum?
4. Is ; bounded above?
Equivalent definitions of supremum
Assume S is an upper bound of the set A.
Which of the following is equivalent to “S is the supremum of A”?
1. If R is an upper bound of A, then S  R .
2. 8R S , R is an upper bound of A.
3. 8R  S , R is not an upper bound of A.
4. 8R < S , R is not an upper bound of A.
5. 8R < S , 9x 2 A such that R < x .
6. 8R < S , 9x 2 A such that R  x .
7. 8R < S , 9x 2 A such that R < x  S .
8. 8R < S , 9x 2 A such that R < x < S .
9. 8" > 0, 9x 2 A such that S " < x .
10. 8" > 0, 9x 2 A such that S " < x  S .
Fix these FALSE statements
1. Let f and g be bounded functions on [a, b]. Then
sup of (f + g )
on [a, b] =
sup of f
on [a, b] +
sup of g
on [a, b]
2. Let a < b < c . Let f be a bounded function on [a, c ]. Then
sup of f
on [a, c ] =
sup of f
on [a, b] +
sup of f
on [b, c ]
3. Let f be a bounded function on [a, b]. Let c 2 R. Then:
sup of (cf )
on [a, b] = c
⇣
sup of f
on [a, b]
⌘
True or False - Suprema and infima
Let A,B ,C ✓ R. Assume C ✓ A. Which statements are true?
If possible, fix the false statements
1. IF A is bounded above, THEN C is bounded above.
2. IF C is bounded below, THEN A is bounded below.
3. IF A and C are bounded above, THEN supC  supA.
4. IF A and C are bounded below, THEN inf C  inf A.
5. IF A and B are bounded, supB  supA, and inf A  inf B ,
THEN B ✓ A.
6. IF A and B are bounded above,
THEN sup(A [ B) = max{supA, supB}.
7. IF A and B are bounded above,
THEN sup(A \ B) = min{supA, supB}.

Warm up: partitions
Which ones are partitions of [0, 2]?
1. [0, 2]
2. {0.5, 1, 1.5}
3. {0, 2}
4. {1, 2}
5. {0, e, 2}
6. {0, 1.5, 1.6, 1.7, 1.8, 1.9, 2}
7.
⇢
n
n + 1
: n 2 N

[ {2}
Warm up: lower and upper sums
Let f (x) = (x 1)2.
Consider the partition P = {0, 1, 3} of the interval [0, 3].
Calculate LP(f ) and UP(f ).
Lower and upper sums of decreasing functions
Let f be a decreasing, bounded function on [a, b].
Let P = {x0, x1, . . . , xN} be a partition of [a, b].
Let xi = xi xi1.
What are LP(f ) and UP(f )?
Joining partitions
Assume
LP(f ) = 2, UP(f ) = 6
LQ(f ) = 3, UQ(f ) = 8
1. Is P ✓ Q?
2. Is Q ✓ P?
3. What can you say about LP[Q(f ) and UP[Q(f )?
The “"–characterization” of integrability
True or False?
Let f be a bounded function on [a, b].
1. IF “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”,
THEN f is integrable on [a, b]
2. IF f is integrable on [a, b]
THEN “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”.
lower sums upper sums
finer partitions finer partitions
I ba (f ) I ba (f )
The “"–characterization” of integrability - Part 1
True or False?
Let f be a bounded function on [a, b].
IF “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”,
THEN f is integrable on [a, b]
Hints:
1. Recall the definition of “f is integrable on [a, b]”.
2. Let P be a partition.
Order the numbers UP(f ), LP(f ), I ba (f ), I
b
a (f ).
(Draw a picture of these numbers in the real line.)
The “"–characterization” of integrability - Part 2
True or False?
Let f be a bounded function on [a, b].
IF f is integrable on [a, b]
THEN “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”.
Hints: Assume f is integrable on [a, b]. Let I be the integral. Fix " > 0.
1. Recall the definition of “f is integrable on [a, b]”.
2. There exist a partition P1 s.t. UP1(f ) < I +
"
2
. Why?
3. There exist a partition P2 s.t. LP2(f ) > I
"
2
. Why?
4. What can you say about UP1(f ) LP2(f )?
5. Construct a partition P s.t. LP2(f )  LP(f )  UP(f )  UP1(f ).
That’s all for today!
Problem set 5 is due on Monday July 11 at 11:59
pm.
David and Lindsey will substitute me for my oce
hours tomorrow. I will be back next week.
Good night!