Calc 1 Lecture Notes
Section 4.2
Page 1 of 5
Section 4.2: Sums and Sigma Notation
Big idea: Sums are an important topic to review in preparation for the geometric concept of an
integral as the area under a curve.
Big skill: You should be able to compute any given sum.
Any sum of numbers where the terms ai follow a pattern that can be represented in terms of
sequential positive integers i can be written in summation notation as:
n
a1  a2  a3  ...  an   ai
i 1
16
 4   i  44.4691966
Example: 1  2  3  2  5 
i 1
On your TI-83 calcualtor:
sum(seq( ( X ), X , 1 ,16,
expression
1 ))
variable begin end increment
You can access the sum and sequence functions by hitting 2nd LIST
Practice:
1. 2 + 4 + 6 + 8 + 10 + … + 20 =
2. 2 + 4 + 8 + 16 + 32 + … + 1024 =
Sums you should know already:
n
Arithmetic Sequence: ai = a + (i – 1)d;
a
i 1
i

n
n
 2a  (n  1)d    a  an  .
2
2
1 rn
Geometric Sequence: ai = ar ;  ai  a
.
1 r
i 1
n
i-1
Calc 1 Lecture Notes
Section 4.2
Page 2 of 5
Summation notation can be used as a shorthand to represent a computation of the
approximate area under a curve.
For example, to compute the approximate area bounded by the curve y  x 2 , the line x = 1, and
the x-axis, we could overlay the area with 5 equal-width rectangles whose height is determined
by the y-value of the graph at the right-hand side of each rectangle:
The area of all the rectangles can then be computed longhand as:
Or written using summation notation as:
In general, the area under any curve given by y  f  x  can be written using summation notation
n
as: A   f  xi  x .
i 1
Notice that if we want to use 10, 20, 100, or more rectangles, it would be nice to have a formula
n
for
i
2
. Specifically, the formula would be nice because then we could compute the area
i 1
2
n
i 1
exactly as lim    
n 
n
i 1  n 
Calc 1 Lecture Notes
Section 4.2
Theorem 2.1: (Some sums of powers of integers)
If n is any positive integer and c is any constant, then
n
i.
 c  c  c  c  c  c  ...  c  cn
i 1
n times
n
ii.
 i  1  2  3  4  5  ...  n 
i 1
n
iii.
n(n  1)
2
2
 i  12  22  32  42  52  ...  n2 
i 1
n
iv.
3
 i  13  23  33  43  53  ...  n3 
i 1
n(n  1)(2n  1)
6
n2 (n  1) 2
4
Theorem 2.2: (A sum of a sum is the sum of the sums…)
For any constants c and d,
n
n
n
i 1
i 1
i 1
  cai  dbi   c ai  d  bi
Practice:
2
 i  1
3. Compute  
 
100
i 1  100 
100
2
n
i 1
4. Compute lim    
n 
n
i 1  n 
Page 3 of 5
Calc 1 Lecture Notes
Section 4.2
Page 4 of 5
Higher order sums of integers:
(This is all “for fun”; it is not in the homework or on the test)
k
n
i
k
i 1
n
nn  1
2
nn  12n  1
6
2
2
n n  1
4
nn  12n  13n 2  3n  1
30
2
2
n n  1 2n 2  2n  1
12
nn  12n  13n 4  6n 3  3n  1
42
2
2
4
n n  1 3n  6n 3  n 2  4n  2 
24
nn  12n  15n 6  15n 5  5n 4  15n 3  n 2  9n  3
90
2
2
6
5
4
n n  1 2n  6n  n  8n 3  n 2  6n  3
20
8
nn  12n  13n  12n 7  8n 6  18n 5  10n 4  24n 3  2n 2  15n  5
66
0
1
2
3
4
5
6
7
8
9
10
General formula for finding higher sums of powers:
n
i
i 1
k

n  1k 1  k11   k11 i  1k 1  i k 1   i k 
n
1
k 1
i 1
Calc 1 Lecture Notes
Section 4.2
Page 5 of 5
Practice:
10
5.
  2i  1
2

i 1
450
6.
i
2
8 
i 9
n
7. Compute a sum of the form
 f  x  x
i 1
i
for f  x   3x  5 ; xi = 0.4, 0.8, 1.2, 1.6, 2.0;
x = 0.4; and n = 5. Then compute the exact area under the curve by taking the limit of the
sum as n  .