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06/18/22 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

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Page 1: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

04/20/23 Math HL1 - Santowski 1

Lesson 4 - Summation Notation & Infinite Geometric Series

Math HL1 - Santowski

Page 2: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

(A) Review

1. Define a “sequence” 2. Define a “series” 3. State 2 formulas associated with arithmetic

sequences and series 4. State 2 formulas associated with geometric

sequences and series

04/20/23 Math HL1 - Santowski 2

Page 3: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

04/20/23 Math HL1 - Santowski 3

(A) Review

A sequence is a set of ordered terms, possibly related by some pattern, which could be defined by some kind of a “formula”

One such pattern is called arithmetic because each pair of consecutive terms has a common difference

A geometric sequence is one in which the consecutive terms differ by a common ratio

Page 4: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

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(B) REVIEW - Formulas 

For an arithmetic sequence, the formula for the general term is:

For an arithmetic sequence then the formula for the sum of its terms is:

Sn=n2u1 +un( ) =

n22u1 + n−1( )d( )

un =u1 +(n−1)d

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(B) REVIEW - Formulas 

For an geometric sequence then the formula for the general term is:

So in general, the formula for the sum of a geometric series is:

 1 1 1( ) ( 1)

1 1

nn

n

u u u rS

r r

un =u1rn−1

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(C) Summation Notation

Summation notation is a shorthand way of saying take the sum of certain terms of a sequence the Greek letter sigma, Σ is used to indicate a summation

In the expression

i represents the index of summation (or term number),

and ai represents the general term of the sequence being summed

So therefore,

1

n

ii

a

1 2 3 41

...n

i ni

a a a a a a

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(D) Ex 1 - Summation Notation

Ex 1 – List the terms of the series defined as below, then determine the sum

6

1

6

1

11ni

ni

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(D) Ex 1 - Summation Notation

Ex 1 – Arithmetic Series

6

1

6

1

6

1

( 1) (1 1) (2 1) (3 1) (4 1) (5 1) (6 1)

( 1) (2) (3) (4) (5) (6) (7)

( 1) 27

n

n

n

n

n

n

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(D) Ex 2 - Summation Notation

Ex 1 – List the terms of the series defined as below, then determine the sum

2 1.1 i

i1

6

∑ 2 1.1 n

n1

6

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(D) Ex 2 - Summation Notation

Ex 1 – Geometric Series

2 1.1( )i

i=1

6

∑ = 2 1.1( )n

n=1

6

2 1.1( )n

n=1

6

∑ =2 1.1( )1+2 1.1( )

2+2 1.1( )

3+2 1.1( )

4+2 1.1( )

5+2 1.1( )

6

2 1.1( )n

n=1

6

∑ =2 1.1( )+2 1.21( )+2 1.331( )+2 1.4641( )+2 1.61051( )+2 1.771561( )

2 1.1( )n

n=1

6

∑ =16.974342

Page 11: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

(E) Examples of Summation Notation

Write out the series expansion for the following and then evaluate the sums:

82

1

10

1

9ln

4

4i

i

i

i

i

i

i

e

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Page 12: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

(F) Word Problems with Geometric Sequences & Series - Annuities An annuity is simplistically a regular deposit made into

an investment plan

For example, I invest $2500 at the beginning of every year for ten years into an account that pays 9% p.a compounded annually at the end of the year.

Determine the terms of the sequence. Determine the total value of my investment at the end of

10 years.

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Page 13: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

(F) Word Problems with Geometric Sequences & Series - Annuities

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(G) Infinite Geometric Series

Show that the sum of n terms of the series 2 + 1 + ½ + ¼ + ..... un is always less than 4, where n is any natural number.

So this will lead us to the idea of “formula” we could use to predict the sum of an infinite number of terms of a geometric series, provided that (or under the condition that …… )

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Page 15: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

(G) Infinite Geometric Series

Explain what the following notations mean:

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S6 = 22−i( )

i=1

6

∑ =

Sn = 22−i( )

i=1

n

∑ =

Sn =limn→ ∞

22−i( )

i=1

n

∑ =

Page 16: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

(F) Examples of Summation Notation

Example – List several terms of the infinite series defined as below, then attempt to determine the sum.

Now explain the what it means to say that a series CONVERGES or a series DIVERGES

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( ) a12nn=1

∑ =

( ) b−1( )

n+1

2nn=1

∑ =

( ) c −2( )n

n=1

∑ =

( ) d 2( )n

n=1

∑ =

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(H) Examples of Infinite Geometric Series

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Find the sum of the following infinite series, if possible. If not, explain why not.

00

00

3

72 (d)

5

110 (c)

75.04 (b) 3

22 (a)

i

i

i

i

i

i

i

i

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(G) Infinite Geometric Series

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Page 19: 10/25/2015 Math HL1 - Santowski 1 Lesson 4 - Summation Notation & Infinite Geometric Series Math HL1 - Santowski

Combination of AS & GS

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Combination of AS & GS

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