Infinite Geometric Sequences and Convergent Series
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Lesson objective: Identify convergent series
and calculate sums to infinity.
The sum of infinite geometric
sequences
Wednesday, 18 September 2024
Wednesday, 18 September 2024
Suppose we have a 2 metre length of string . . .
. . . which we cut in half
We leave one half alone and cut the 2
nd
in half again
. . . and again cut the last piece in half
Geometric Series
2 m
1 m
1 m
1 m
1 m
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.
Geometric Series
Even though there are an infinite number of terms, this
series
converges
to 2.
Geometric Series
Number of terms,
n
Sum,
S
n
We will find a formula for the sum of an infinite number
of terms of a Geometric series. This is called “the sum
to infinity”,
S
e.g. For the Geometric Series
This term gets smaller as
n
gets larger.
we know that the sum of
n
terms is given by
u
1
S
n
=
(1
– r
n
)
(1
– r
)
u
1
= 1
Sum to infinity
We write:
u
1
S
n
=
(1
– r
n
)
1
– r
u
1
=
1
– r
= 2
u
1
S
∞
=
1
– r
So, the formula for the infinite geometric series is
n → ∞
Sum to infinity
However, if, for example
r =
2
,
There is no sum to infinity
Divergent series
r
n
=
2
n
Also, if
r
1
, ( e.g.
r
=
2
)
,
If
r
is
any
value greater than
1
, the series diverges.
So, again the series diverges.
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
, . . .
Convergent Series
r
n
±
as
n
⇒
r
≠
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.
so
r
does satisfy
1 <
r
< 1
The series converges to
27
18 + 6 + 2 + ….,
u
1
= 18
Solution:
Convergent Series
Example 1
S
= 27
r =
the series converges.
Use the sum of infinite series to write 0.55555... as a
rational number
0.55555….
=
u
1
=
Solution:
Convergent Series
Example 2
S
=
r =
+
+
+
+
…
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.
Solution:
so
r
does satisfy
1 <
r
< 1
The series converges to
1.6
Convergent Series
Example 3
u
1
=
2
r =
the series converges.
=
1.6
Convergent Series
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Explore the concept of infinite geometric sequences in mathematics through the example of cutting a string into halves. Learn how to identify convergent series and calculate sums to infinity, distinguishing between convergent and divergent series based on the common ratio. Delve into the formula for finding the sum of an infinite number of terms in a geometric series, understanding how the sum approaches zero as the number of terms increases or when the common ratio is other than 1.
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Presentation Transcript
Wednesday, 18 September 2024 The sum of infinite geometric sequences Lesson objective: Identify convergent series and calculate sums to infinity. www.mathssupport.org
Geometric Series Suppose we have a 2 metre length of string . . . 2 m . . . which we cut in half 1 m 1 m We leave one half alone and cut the 2nd in half again 1 2 m 1 2 m 1 m . . . and again cut the last piece in half 1 2 m 1 4 m 1 4 m 1 m www.mathssupport.org www.mathssupport.org
Geometric Series Continuing to cut the end piece in half, we would have in total ? +1 2+1 4+1 8+ ... 1 1 1 m 1 m m m 2 4 8 In theory, we could continue for ever, but the total length would still be 2 metres, so 1 +1 2+1 4+1 8+ ... = 2 This is an example of an infinite series. www.mathssupport.org www.mathssupport.org
Geometric Series 1 +1 2+1 4+1 The series 8+ ... = 2 r =1 is a Geometric Series with the common ratio 2. . Even though there are an infinite number of terms, this series converges to 2. 2 Sum, Sn 1 4 5 6 2 1 3 Number of terms, n www.mathssupport.org www.mathssupport.org
Sum to infinity We will find a formula for the sum of an infinite number of terms of a Geometric series. This is called the sum to infinity , S e.g. For the Geometric Series 1 +1 2+1 4+1 8+ ... = 2 we know that the sum of n terms is given by ? 1 2 u1 = 1 r=1 1 1 (1 rn) u1 (1 r) Sn= ??= 1 1 2 2 ?. As n varies, the only part that changes is 1 2 This term gets smaller as n gets larger. www.mathssupport.org www.mathssupport.org
Sum to infinity ? approaches zero. As n approachesinfinity, 1 2 We write: ? 0 1 2 As n , 0 u1 (1 rn) 1 r u1 So, for r=1 Sn= = 2, 1 r 1 For the series 1 +1 2+1 4+1 = 2 8+ ... ? = 1 1 2 So, the formula for the infinite geometric series is u1 S = 1 r n www.mathssupport.org www.mathssupport.org
Divergent series However, if, for example r = 2, rn = 2n n As n increases, also increases. In fact, 2 As n , ?? The geometric series with diverges = = r 2 There is no sum to infinity www.mathssupport.org www.mathssupport.org
Convergent Series If r is any value greater than 1, the series diverges. Also, if r 1, ( e.g.r = 2), So, again the series diverges. 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, . . . rn as n A Geometric Series converges only if the common ratio r lies between 1 and 1. ?1 for 1 < r < 1, r ? = 1 ? This can also be written as | r | < 1 www.mathssupport.org www.mathssupport.org
Convergent Series Example 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. 18 + 6 + 2 + ., u1 = 18 6 18 ?1 Solution: ? = 1 ? 18 =1 r = 3 ? = 1 3 1 so r does satisfy 1 < r < 1 the series converges. S = 27 The series converges to 27 www.mathssupport.org www.mathssupport.org
Convergent Series Example 2 Use the sum of infinite series to write 0.55555... as a rational number 5 10 5 10 5 5 5 0.55555 . = + + + 10000+ 100 1000 ?1 u1 = Solution: ? = 1 ? 5 10 1 10 r = ? = 1 10 1 5 9 S = www.mathssupport.org
Convergent Series Convergent Series Example 3 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. 2 8 2 Solution: u1 = 2 1 2 2 4 1 1 1 + + + + . . . 32 = 1 so r does satisfy 1 < r < 1 the series converges. 2 1 4 r = ?1 S =8 5=1.6 ? = ? = 1 ? 1 The series converges to 1.6 www.mathssupport.org
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org www.mathssupport.org
