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Sum Of A Geometric Series

11 x1 t14 06 sum of a geometric series (2012)

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Page 1: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series

Page 2: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series 12 n

n arararaS

Page 3: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series 12 n

n arararaS nn

n arararararrS 132

Page 4: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series 12 n

n arararaS nn

n arararararrS 132

aarSr n

n 1

Page 5: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series 12 n

n arararaS nn

n arararararrS 132

aarSr n

n 1

1

1

r

raS

n

n

Page 6: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series 12 n

n arararaS nn

n arararararrS 132

aarSr n

n 1

1

1

r

raS

n

n 1 if , r

Page 7: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series 12 n

n arararaS nn

n arararararrS 132

aarSr n

n 1

1

1

r

raS

n

n

OR

1 if , r

r

raS

n

n

1

1

Page 8: 11 x1 t14 06 sum of a geometric series (2012)

Sum Of A Geometric

Series 12 n

n arararaS nn

n arararararrS 132

aarSr n

n 1

1

1

r

raS

n

n

OR

1 if , r

r

raS

n

n

1

1 1 if, r

Page 9: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum)

Page 10: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

Page 11: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

Page 12: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

Page 13: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

r

a

1

Page 14: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

r

a

1

1 if ,1

rr

aS

Page 15: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

r

a

1

1 if ,1

rr

aS

1862 of terms10first theof sum theFind .. ige

Page 16: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

r

a

1

1 if ,1

rr

aS

1862 of terms10first theof sum theFind .. ige

10 and 3,2 nra

Page 17: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

r

a

1

1 if ,1

rr

aS

1862 of terms10first theof sum theFind .. ige

10 and 3,2 nra 1

1

r

raS

n

n

Page 18: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

r

a

1

1 if ,1

rr

aS

1862 of terms10first theof sum theFind .. ige

10 and 3,2 nra 1

1

r

raS

n

n

13

132 10

10

S

Page 19: 11 x1 t14 06 sum of a geometric series (2012)

Sum To Infinity (Limiting Sum) 1: rNOTE

0lim ,1 If

n

nrr

r

raS

n

nn

n

1

1limlim

r

a

1

1 if ,1

rr

aS

1862 of terms10first theof sum theFind .. ige

10 and 3,2 nra 1

1

r

raS

n

n

13

132 10

10

S

59048

Page 20: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii

Page 21: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

Page 22: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

2

3

Page 23: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

2

3

6 ,2

1 nr

Page 24: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

r

raS

n

n

1

12

3

6 ,2

1 nr

Page 25: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

r

raS

n

n

1

12

3

6 ,2

1 nr

2

11

2

11

2

36

6

S

Page 26: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

r

raS

n

n

1

12

3

6 ,2

1 nr

2

11

2

11

2

36

6

S

64

189

1

2

64

63

2

3

Page 27: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

r

raS

n

n

1

1

sum? limiting a have 7

2456 Does iii

2

3

6 ,2

1 nr

2

11

2

11

2

36

6

S

64

189

1

2

64

63

2

3

Page 28: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

r

raS

n

n

1

1

sum? limiting a have 7

2456 Does iii

156

4r

2

3

6 ,2

1 nr

2

11

2

11

2

36

6

S

64

189

1

2

64

63

2

3

Page 29: 11 x1 t14 06 sum of a geometric series (2012)

8

3

1

2

16

n

n

ii2

2

16

a

r

raS

n

n

1

1

sum? limiting a have 7

2456 Does iii

156

4r

sum limiting a hasit ,1 as r

2

3

6 ,2

1 nr

2

11

2

11

2

36

6

S

64

189

1

2

64

63

2

3

Page 30: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

Page 31: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

Page 32: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

r

aS

1

Page 33: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

r

aS

1

1

2

11

2

1

Page 34: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

fraction a as 630. Write v

r

aS

1

1

2

11

2

1

Page 35: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

fraction a as 630. Write v

000036.00036.036.063.0

r

aS

1

1

2

11

2

1

Page 36: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

fraction a as 630. Write v

000036.00036.036.063.0

01.0 ,36.0 ra

r

aS

1

1

2

11

2

1

Page 37: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

fraction a as 630. Write v

000036.00036.036.063.0

01.0 ,36.0 ra

r

aS

1

r

aS

1

1

2

11

2

1

Page 38: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

fraction a as 630. Write v

000036.00036.036.063.0

01.0 ,36.0 ra

r

aS

1

r

aS

1

1

2

11

2

1

11

4

99

36

01.01

36.0

Page 39: 11 x1 t14 06 sum of a geometric series (2012)

8

1

4

1

2

1 iv

2

1,

2

1 ra

fraction a as 630. Write v

000036.00036.036.063.0

01.0 ,36.0 ra

r

aS

1

r

aS

1

1

2

11

2

1

11

4

99

36

01.01

36.0

Exercise 6J; 3cf, 5, 7b

8a(i), 9, 12, 14, 17b

Exercise 6K; 2adgj, 3b,

4a, 6bf, 8, 11a, 17ace, 19*

Exercise 6L, 1dg, 2d, 4a