Binomial Problems © Christine Crisp “Teach A Level Maths” Statistics 1.
33: Geometric Series Part 2 Sum to Infinity © Christine Crisp “Teach A Level Maths” Vol. 1: AS...
18
33: Geometric Series 33: Geometric Series Part 2 Part 2 Sum to Infinity Sum to Infinity © Christine Crisp “ “ Teach A Level Maths” Teach A Level Maths” Vol. 1: AS Core Vol. 1: AS Core Modules Modules
-
Upload
buddy-elliott -
Category
Documents
-
view
229 -
download
11
Transcript of 33: Geometric Series Part 2 Sum to Infinity © Christine Crisp “Teach A Level Maths” Vol. 1: AS...
33: Geometric Series33: Geometric SeriesPart 2Part 2
Sum to InfinitySum to Infinity
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Geometric series – Sum to Infinity
Module C2
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
Geometric series – Sum to Infinity
Suppose we have a 2 metre length of string . . .
. . . which we cut in half
We leave one half alone and cut the 2nd in half again
m 1 m 1
m 1 m 21
. . . and again cut the last piece in half
m 1 m 21
m 41 m
41
m 21
Geometric series – Sum to Infinity
Continuing to cut the end piece in half, we would have in total
In theory, we could continue for ever, but the total length would still be 2 metres, so
This is an example of an infinite series.
m 1 m 21
...181
41
21
m 41 m
81
2...181
41
21
Geometric series – Sum to Infinity
Even though there are an infinite number of terms, this series converges to 2.
The seriesis a G.P. with the common ratio . 2
1r
2...181
41
21
Number of terms, n
Sum
nS
Geometric series – Sum to Infinity
We will find a formula for the sum of an infinite number of terms of a G.P. This is called “the sum to infinity”, Se.g. For the
G.P....1
81
41
21
2121
1
11
n
nS
As n varies, the only part that changes is . n21
r
raS
n
n
1
)1(
This term gets smaller as n gets
larger.
we know that the sum of n terms is given by
Geometric series – Sum to Infinity
We write: 0,
21 nn As
As n approaches infinity, approaches
zero. n
21
r
raS
n
n
1
)1(So, for ,
21r
r
aS
1
0
...1 81
41
21 For the
series
21
1
21
S
Geometric series – Sum to Infinity
nnr 2
However, if, for example r = 2,
As n increases, also increases. In fact, n2
nn 2, AsThe geometric series with diverges
2rThere is no sum to infinity
Geometric series – Sum to Infinity
r
aS
1
11 rfor
Convergence
Also, if r 1, ( e.g. r = 2 ), nr n as
If r is any value greater than 1, the series diverges.
So, again the series diverges.
A Geometric Series converges only if the common ratio r lies between 1 and 1.
If r = 1, all the terms are the same.If r = -1, the terms have the same magnitude but they alternate in sign. e.g. 2, -2, 2, -2, . . .
This can also be written as 1r
Geometric series – Sum to Infinity
e.g. 1. For the following geometric series,
write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity.
...2321
81
21
411
2
S
r
aS
1
Solution: 4
1
221
rso r does satisfy 1 < r <
1
6158 or S
The series converges to 61
Geometric series – Sum to Infinity
SUMMARY
r
raS
n
n
1
)1(
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is 1 nn aru
...,,,, 32 ararara
The sum of n terms
is
1
)1(
r
raS
n
no
r The sum to infinity
is
11 r 1r;1 r
aS
or
Geometric series – Sum to Infinity
Exercises1. For the following geometric series, write
down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity.
...32)(4
2729 a
...13)(91
31 b
Ans: (a) so the series diverges.23r
r
aS
1
(b) so the series converges.3
1r
2524
9
1
3
31
or S
Geometric series – Sum to Infinity
Geometric series – Sum to Infinity
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
Geometric series – Sum to InfinityWe will find a formula for the sum of an infinite
number of terms of a G.P. This is called “the sum to infinity”, Se.g. For the
G.P....1
81
41
21
2121
1
11
n
nS
As n varies, the only part that changes is . n21
r
raS
n
n
1
)1(
We write:
0,21 nn As
This term gets smaller as n gets
larger.As n approaches infinity, approaches
zero. n21
we know that the sum of n terms is given by
Geometric series – Sum to Infinity
r
raS
n
n
1
)1(
nnr 2
So, for 21r
r
aS
1
However, if, for example r = 2,
As n increases, also increases. In fact, n2
nn 2, AsThe geometric series with diverges
2r
21
1,...1
218
141
21
SFor the
series
There is no sum to infinity
0,21 nn As
Geometric series – Sum to InfinityConvergen
ce
Also, if r 1, ( e.g. r = 2 ), nr n as
If r is any value greater than 1, the series diverges.
So, again the series diverges.
A Geometric Series converges only if the common ratio r lies between 1 and 1.
r
aS
1
11 rfor
If r = 1, all the terms are the same.If r = -1, the terms have the same magnitude but they alternate in sign. e.g. 2, -2, 2, -2, . . .
1r ( or )
Geometric series – Sum to Infinity
SUMMARY
r
raS
n
n
1
)1(
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is 1 nn aru
...,,,, 32 ararara
The sum of n terms
is
1
)1(
r
raS
n
no
r The sum to infinity
is
11 r 1r;1 r
aS
or
