The sum of the infinite and finite geometric sequence

The sum of the infinite and finite geometric sequence

2

The sum of the first n terms of a sequence is represented by summation notation.

index of summation

upper limit of summation

lower limit of summation

The sum of a finite geometric sequence is given by

5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?

n = 8a1 = 5

The sum of the terms of an infinite geometric sequence is called a geometric series.

a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .

If |r| < 1, then the infinite geometric series

has the sum

Example: Find the sum of

The sum of the series is

Convergent and Divergent Series

Convergent and Divergent Series

If the infinite series has a sum, or limit, the series is convergent.

If the series is not convergent, it is divergent.

Ways To Determine Convergence/Divergence

1. Arithmetic – since no sum exists, it diverges

2. Geometric: If |r| > 1, diverges If |r| < 1, converges since the sum

exists3. Ratio Test (discussed in a few

minutes)

Example

Determine whether each arithmetic or geometric series is convergent or divergent.

1/8 + 3/20 + 9/50 + 27/125 + . . . r=6/5 |r|>1 divergent

18.75+17.50+16.25+15.00+ . . . Arithmetic series divergent

65 + 13 + 2 3/5 + 13/25 . . . r=1/5 |r|<1 convergent

Other Series

When a series is neither arithmetic or geometric, it is more difficult to determine whether the series is convergent or divergent.

Ratio TestIn the ratio test, we will use a ratio of an and an+1 to determine the convergence or divergence of a series.

1

nIf lim 1, the absoconverg lutelyesn

n

n

aaa

1

nIf lim 1, th diverges.e n

nn

aaa

1

nIf lim 1, the ratio test i incos nclusive.n

n

aa

Review:2

2nlim n ba

ndn ce

2

3n0lim an bn c

dn e

Leading coefficient is a

Leading coefficient is

d

Denominator degree is greater

ad

Test for convergence or divergence of:

23

n

na n

1

23

n

nn

1

12( 1)3n

n

na

1

1

2( 1)

23

3

nn

n

n

naa

n

11 23

n nnn

1 23

nn

1lim n

nn

aa

2 1lim3 n

nn

23

Since this ratio is less than 1, the series converges.

Test for convergence or divergence of:

2

2n n

na 2

11

( 1)2nn

na

2

2

11

( 1)2

2

nn

nn

na

na

2

1 2

1 22

n

n

nn

2

2 1

( 1) 22

n

n

nn

1lim n

nn

aa

Since this ratio is less than 1, the series converges.

2

1 2nn

n

2

2

1 2 12n n

n

2

2

1 2 1lim2 n

nnn

The ratio of the leading coefficients is

112

Test for convergence or divergence of:

( 2)( 1)n

nan n

1

( 3)( 1)( 2)n

nn

an

1

( 3)( 1)( 2)

( 2)( 1)

n

n

nn nanan n

( 3) ( 1)( 1)( 2) ( 2)

n n nn n n

1lim n

nn

aa

Since this ratio is 1, the test is inconclusive.

Coefficient of n2 is 1

1

1

( 1) ( 2)( 1)

n

n

nn n

2

( 3)( 2)n nn

2

( 3)lim( 2)n

n nn

Coefficient of n2 is 1

1

Example

Use the ratio test to determine if the series is convergent or divergent.

1/2 + 2/4 + 3/8 + 4/16 + . . .

1 1

12 2

_ _n nn n

n na and a

Since r<1, the series is convergent.

1

1

11 2 12lim lim lim 1/ 2

2 22

nn

nn n nn

nn nr

n n n

Example

Use the ratio test to determine if the series is convergent or divergent.

1/2 + 2/3 + 3/4 + 4/5 + . . .

1_ 1 11 ( 1) 2

_1n n

n n na and an n n

Since r=1, the ratio test provides no information.

2

2

1 1 1 2 12lim lim lim 12 21

n n n

n n n n nnr n n n n nn

Example

Use the ratio test to determine if the series is convergent or divergent.

2 + 3/2 + 4/3 + 5/4 + . . .

11 ( 1) 1

1_ _ 2

1n nn n na and an n n

Since r=1, the ratio test provides no information.

2

2

2 2 21lim lim lim 11 1 1 2 1n n n

n n n n nnr n n n n nn

Example

Use the ratio test to determine if the series is convergent or divergent.

3/4 + 4/16 + 5/64 + 6/256 + . . .

12 2( 1) 2 2_ _2 ( 2) 1 32 2 2n nn n n

n n na and a

Since r<1, the series is convergent.

22 2

2 2 22

33 2 3 12lim lim lim2 2 2 2 ( 2) 4

2

nn

nn n nn

nn nr

n n n

Example

Use the ratio test to determine if the series is convergent or divergent.

1_ _1 1! ( 1)!n na and an n

Since r<1, the series is convergent.

11 ! 1( 1)!lim lim lim 01 ( 1)! 1 1!

n n n

nnrn nn

1 1 11 ....1 2 1 2 3 1 2 3 4

Example

Use the ratio test to determine if the series is convergent or divergent.

1

12 2

( 1) ( 1)(_

2_

)

n n

n na and an n n n

Since r>1, the series is divergent.

112

2 ( 1) 2( 1)( 2)lim lim lim 22 ( 1)( 2) 2 2

( 1)

nn

n nn n n

n n nn nrn n n

n n

2 4 8 16 ....1 2 2 3 3 4 4 5