MATH 2413 - Calculus I
Sigma Notation/Summation Exercises
Some Relevant Formulas and Properties
n
n
∑
kf ( j) =k
j =m
∑
n
n
∑ [ f ( j) ± g( j)] = ∑
f ( j)
j =m
j =m
n
j =m
n
n(n + 1)
∑ j= 2
j =1
n
∑
f ( j) ±
n
g( j)
j =m
∑ c = nc
j =1
n
n(n + 1)(2n + 1)
∑j =
6
j =1
n(n + 1)
∑ j =  2 
j =1
2
2
3
In Exercises 1-16, evaluate the given summation.
3
∑2
1.
3
j
2.
j =0
6.
10.
j =1
500
(−1) j
∑ j
j =1
∑(
∑j
)
j =1
90
14.
∑(
j =1
∑ (−1) j
4.
j =3
5
7.
∑3
1000
8.
j =1
11.
∑ 5j
100
12.
j =1
)
−2 j 2 + 3 j − 5
∑ (2 j − 3) 2
j =1
∑ −7 j 2
j =1
100
15.
∑7
j =1
1000
2
j =1
3j 2 − 5 j + 2
6
j =1
50
∑j
13.
∑ (2 j − 3)
4
24
∑ j!
j =0
100
9.
3.
j =0
4
5.
∑j
5
2
100
16.
∑ (− j + 2)2
j =1
In Exercises 17-20, express each summation in a simplified algebraic expression that does not
use sigma notation.
n
n
n
n

1 

2
2 1 
17. ∑ (3j − 2)
18. ∑ (3j − 4)
19. ∑ ( 3j − 5) ⋅ 2 
20. ∑ ( j − 2) ⋅ 3 
n 
n 
j =1
j =1
j =1 
j =1 
€In Exercises 21-22, evaluate the given expression.
n
n

1 

2 1 
21. lim ∑ (3 j − 5) ⋅ 2 
22. lim ∑ ( j − 2) ⋅ 3 
n→∞ j =1 
n 
n→∞ j =1 
n 
€
23. Find the exact area under f (x) = 3x + 5 from a=0 to b=3 by using an infinite summation.
24. Find the exact area under f (x) = 2x 2 + 8 from a=1 to b=3 by using an infinite summation.
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(Thomason - Fall 2003)
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