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Pre-Calculus

Sigma Notation and Series

Arithmetic series

Precalculus Unit 8
DEFINITION: A Series is the sum of the terms of a sequence.
Sigma Notation or (Summation Notation)
Σαπ
n=1
(b)
Ex 1) Find the sum of each finite sequence.
(c)
5
1
iMu i Mu
n=1
n=3
10
Σ
n=5
"n² 1+4+9+16+25
The variable below sigma (n in this case) is called the "index"
The number below sigma (1 in this case) is which term begins the series, called the "lower bound"
The number above sigma (k in this case) is which term ends the series, called the "upper bound"
The expression to the right of sigma (an in this case) is the explicit formula used to generate the
terms of the series.
n
n
T
=
=
1
2²
Notes-Sigma Notation and Series
+
2
Gn
n=1
4
+
6
(d)
Σ² = 2+2+2+2+2+2=12
n=1
5
5+6+7+8+9+10=45
Ex2) Write the following sums using sigma notation:
a) 1+4+9+16+25+36+49 +64 +81
d) 6+2-2-6-10-14-18-n.
This symbol means "Add" → It's called "Sigma"
47
d=-4
(-4n+10) 4₁ = 6
an=6+-4(n-1)
55
Ex3) Find each sum.
a) 3+7+11+15+19+23+27
7 (3+27) = 105)
2
b) 2+4+6+8+10+12
Zan
n=1
an=2n
e) 729 +243 +81+27+9+3
729 (1₁-1
n=1
FINITE ARITHMETIC SERIES S₂ = a₁ = 2(a₁ + a₂)
k=l
arith.
d=2
a₁ = 2
On=2+2(n-1) n=1
n=# terms
9₁ = 1st term
An = last term
c) 625+125+ 25 +...
East+
b) 1+3+5+7+...+49
an = a₁ + d(n-1)
49 = 1+2(n-1)
48=2(n-1)
34= n-1 n=25
n-i geom
r===
a₁-E
n=2
f) 8+27 +64 +125
an=660
3
25/ (1+49) = 620 Ex4) A corner section of a stadium has 8 seats along the front row. Each successive row has 2 more seats than
the row preceding it. If the top row has 24 seats how many seats are in the entire section?
8+10+12+
+24
Sum of a Finite
Geometric Series:
S₁₁ = a₁
1-p"
1-r
Ex5) Find the sum of each geometric series.
n-1
»Σ (1)
a) 4
3
4
V = 0.15
3
1-0.75
F=
2
24 = 8+2(n-1)
16=2(n-1)
8= n-i
9=n
1-( - ))
4 + - ²4/12 + 4 + + 4
3
6561
T-(-3)
it
9
2
*
Ex6) Determine if each of the following series converges. If the series converges, find the sum.
5
-4
a) [3(0.75)-
b) Ž
n=0
/ 0,75/=.75 ²1
+
19684
6561
2
Series Converges
12
+
✓
""
4
|=| >1
Series diverges
no Sum
TT
8 t
IT
9
2 (8+24) = (144 seats
!
+...
Sum of an Infinite
Geometric Series:
Σαι
•p.1-1
if & only if |r|<1
k=1
If an infinite series has a sum it is convergent.
If it does not, it is divergent
b) 5 + 15 +45 +...+98415
98415= 5(3)
19683=3"-1
In 19683= (n-1) In3
9=n-1
10=n
5-1-(3)
1-3
147620
-√²/2+|+ 1/² + 1/5 +₁1,
r = - + 1/² | - ² 1² | = 1²1 1²2 < 1 ✓
|
시
- 1/4
78-3
-25
1- (- 4/12)
36
-1
d) Σ
2 (19) ™¹ = 1 + ²/2 + + + + + + + ...
10
r == 1/2/2 | 2² | == 1 1/2 2 ² 1 ✓
L
Series Converges
1
T-½ 2
2
6

Sigma Notation and Series

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Sigma Notation and Series

Similar Content

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying