term and the sum of the first n terms
Consider the
concept of convergence and divergenceThe formula for the sum to infinity of a GP
Разделы презентаций
- Разное
- Английский язык
- Астрономия
- Алгебра
- Биология
- География
- Геометрия
- Детские презентации
- Информатика
- История
- Литература
- Математика
- Медицина
- Менеджмент
- Музыка
- МХК
- Немецкий язык
- ОБЖ
- Обществознание
- Окружающий мир
- Педагогика
- Русский язык
- Технология
- Физика
- Философия
- Химия
- Шаблоны, картинки для презентаций
- Экология
- Экономика
- Юриспруденция
04 Series
Содержание
- 1. 04 Series
- 2. Today’s ObjectivesDefinition of a Geometric Progression (GP)Formulae
- 3. Geometric Progressions (GPs)A geometric progression is a
- 4. General term of a GPThe general formula
- 5. Sum of the 1st n terms of
- 6. We consider the same expression multiplied by
- 7. Hence,If we do rSn – Sn instead
- 8. Example 1The 5th term of a GP
- 9. Example 1 solutionSince S10 > 0 and
- 10. Example 1 solutionUsing a and r in the formula for the sum to n:
- 11. Example 2The sum of the first n
- 12. Example 2 solutionWe need to find Tn and Tn – 1 and we are given Sn:
- 13. Example 2 solutionNow consider whether Tn/Tn –
- 14. Example 2 solutionFinally, finding the sum of
- 15. Example 3A prize fund is set up
- 16. Example 3 solutionLet Pn is the amount
- 17. Example 3 solutionSo, at the end of
- 18. Example 3 solutionContinuing this pattern, at the end of the nth year and simplifying:
- 19. Example 3 solutionTherefore:Now Pn is the amount
- 20. Example 3 solutionSo,
- 21. ConvergenceConsider the series where each term is
- 22. ConvergenceWe can see that for this series
- 23. DivergenceConsider the series where each term is double the preceding term, i.e. Tn = 2Tn-1.
- 24. DivergenceWe can see that for this series
- 25. DivergenceConsider the series where we add 10
- 26. DivergenceWe can see that for this series
- 27. Convergence and divergenceSo we can generalise that
- 28. Sum to infinity of APsSince, for an
- 29. Sum to infinity of GPsThe first two
- 30. Sum to infinity of GPsConsider the formula
- 31. Sum to infinity of GPsFor |r| < 1, i.e. – 1 < r < 1:
- 32. Sum to infinity of GPsFor |r| >
- 33. Sum to infinity of a GP |r|
- 34. Example 4Determine if the following series converge
- 35. Example 4 Solution3 + 5 + 7
- 36. Example 4 Solution3 + 9/2 + 27/4 + ...
- 37. Example 5Find the condition on x so
- 38. Example 5 SolutionFirst we need to establish what type of series it is:
- 39. Example 5 SolutionThe series will converge if |r| < 1:
- 40. Example 5 SolutionTo evaluate the series when
- 41. Example 5 SolutionSo,
- 42. Example 6Express the recurring decimal 0.15̅7̅6̅ as
- 43. Example 6 SolutionSo,
- 44. Example 6 SolutionNow consider the term in the bracket:
- 45. Example 6 SolutionSimplifying:
- 46. Arithmetic meanIf 3 numbers, say p1, p2
- 47. Geometric meanIf 3 numbers, say p1, p2
- 48. Example 7The 3rd term of a convergent
- 49. Example 7 SolutionSo,
- 50. Example 7 SolutionNow simplify and consider that the series is convergent:
- 51. Example 7 SolutionNow we know r and
- 52. In summaryThe general term of a GP
- 53. In summaryA series may be the sum
- 54. In summaryA series is convergent if its
- 55. In summaryThe arithmetic mean of x and y is:The geometric mean of x and y is:
- 56. Скачать презентанцию
Today’s ObjectivesDefinition of a Geometric Progression (GP)Formulae for the nth term and the sum of the first n termsConsider the concept of convergence and divergenceThe formula for the sum to infinity
- ВКонтакте
- РћРТвЂВВВВВВВВнокласснРСвЂВВВВВВВВРєРСвЂВВВВВВВВ
- РњРѕР№ Р В Р’В Р РЋРЎв„ўР В Р’В Р РЋРІР‚ВВВВВВВВРЎР‚
Слайды и текст этой презентации
Слайд 2Today’s Objectives
Definition of a Geometric Progression (GP)
Formulae for the nth
term and the sum of the first n terms
concept of convergence and divergenceThe formula for the sum to infinity of a GP
Слайд 3Geometric Progressions (GPs)
A geometric progression is a sequence where each
term is a constant multiple of the preceding term.
The
constant multiple is known as the common ratio and we use the letter r. It can take any real value.E.g. 12, 6, 3, 1.5, 0.75, …
This is a GP with first term 12 and common ratio of 0.5
Слайд 4General term of a GP
The general formula for a term
in a GP is:
Tn = arn-1
where a is the first
term, r is the common ratio and n is the position number
Слайд 5Sum of the 1st n terms of a GP
Considering the
sum of the first n terms of the sequence (i.e.
the series), where a is the first term, r is the common ratio:But as with the sum of an AP this is rather long so we use a similar approach to simplify this formula.
Слайд 6We consider the same expression multiplied by r and then
subtract one from the other:
Sum of the 1st n
terms of a GP
Слайд 7Hence,
If we do rSn – Sn instead we get,
Use the
first if r < 1 and use the second if
r > 1Sum of the 1st n terms of a GP
Слайд 8Example 1
The 5th term of a GP is 8, the
3rd term is 4, and the sum of the first
ten terms is positive. Find the 1st term, the common ratio, and the sum of the first ten terms.We will find the 1st term and common ratio simultaneously using the general term Tn = arn – 1. Then we will use the formula for Sn.
Слайд 11Example 2
The sum of the first n terms of a
series is 3n – 1. Show that the terms in
this series are in geometric progression and state the common ratio and the first term and the sum of the second set of n terms in this series.If terms are in GP then there will be a constant common ratio and therefore Tn/Tn – 1 will be a constant and will be the common ratio. We can use Tn to state the first term. The sum of the second set on n terms will be given by S2n – Sn.
Слайд 13Example 2 solution
Now consider whether Tn/Tn – 1 is a
constant:
It is a constant and therefore the series is in
GP and the common ratio is 3.Now comparing our Tn = 3n – 1(2) with the generalised Tn = arn – 1 and since r = 3 we conclude that the 1st term a = 2.
Слайд 14Example 2 solution
Finally, finding the sum of the second set
of n terms:
This method can be used to find the
sum of any consecutive terms for an AP or GP, e.g. S8 – S4 = T5 + T6 + T7 + T8
Слайд 15Example 3
A prize fund is set up with a single
investment of £2000 to provide an annual prize of £150.
The fund accrues interest at 5% per year paid at the end of the year. If the first prize is awarded one year after the investment, find the number of years for which the full prize can be awarded.We need to find an expression for the value of the prize fund after n years. We then use this formula to find the required value of n.
Слайд 16Example 3 solution
Let Pn is the amount in the prize
fund at the end of n years. Then at the
end of the first year:
Слайд 17Example 3 solution
So, at the end of the second year:
Using
the same method, at the end of the third year:
Слайд 19Example 3 solution
Therefore:
Now Pn is the amount in the prize
fund after the prize has been taken out. So we
want to know the least value of n when Pn > 0 since if Pn > 0 then there was enough in the prize fund in the nth year.
Слайд 21Convergence
Consider the series where each term is half of the
preceding term, i.e. Tn = 0.5Tn-1.
Слайд 22Convergence
We can see that for this series the sum of
the series converges to the value of 1.
This series is
therefore convergent.This series is actually a GP with first term a = 0.5 and common ratio r = 0.5.
Слайд 23Divergence
Consider the series where each term is double the preceding
term, i.e. Tn = 2Tn-1.
Слайд 24Divergence
We can see that for this series the sum of
the series continual increases and does not converge.
This series is
therefore divergent.This series is actually a GP with first term a = 0.5 and common ratio r = 2.
Слайд 26Divergence
We can see that for this series the sum of
the series continual increases and does not converge.
This series is
therefore divergent.This series is actually an AP with first term a = 0.5 and common difference d = 10.
Слайд 27Convergence and divergence
So we can generalise that all APs are
divergent.
Also for GPs,
If |r| > 1 then the GP
is divergentIf |r| < 1 then the GP is convergent
Слайд 28Sum to infinity of APs
Since, for an AP, we are
continually adding or continually subtracting, as n increases so the
sum to n increases or decreases depending on whether the common difference is positive or negative respectively.So for an AP when n → ∞,
S∞ → ∞ when d > 0
S∞ → –∞ when d < 0
It is therefore not possible to find a formula for the sum to infinity for an AP.
Слайд 29Sum to infinity of GPs
The first two examples above were
GPs but the first was convergent and the second was
divergent.For a divergent GP, i.e. |r|>1, then the sum to infinity will be infinite and so we can not find a formula for the sum to infinity.
However for a convergent GP, i.e.|r|<1, then it is possible to find a formula for the sum to infinity.
Слайд 30Sum to infinity of GPs
Consider the formula for the sum
of the first n terms of a GP:
Now consider the
value of rn as n → ∞ for |r| < 1 and for |r| > 1
Слайд 34Example 4
Determine if the following series converge or diverge and
if possible find the sum to infinity:
3 + 5 +
7 + ...1 – ¼ + 1/16 – 1/64 + ...
3 + 9/2 + 27/4 + ...
To determine if a series converges or diverges, first establish if it is an AP or a GP then, if it is a GP, find the common ratio. Only if it is a GP with |r|<1, i.e. Convergent, can we find the sum to infinity.
Слайд 37Example 5
Find the condition on x so that the following
series converges:
Now evaluate the series when x = 1.5.
Слайд 40Example 5 Solution
To evaluate the series when x = 1.5
consider whether this value is within the range of convergence,
i.e. is x = 1.5 in the range – 1 < x < 3?Since it is in the range of convergence and since we have not been told how many terms we must consider, we can use the formula for the sum to infinity.
Слайд 42Example 6
Express the recurring decimal 0.15̅7̅6̅ as a fraction in
its lowest terms.
We can consider this recurring decimal as a
sum and therefore as a series. This will allow us to convert it to a fraction.
Слайд 46Arithmetic mean
If 3 numbers, say p1, p2 and p3, are
in arithmetic progression then the middle number p2 is the
arithmetic mean of p1 and p3.Since if p1 = a then p2 = a + d and p3 = a + 2d then:
Hence the arithmetic mean of x and y is:
Слайд 47Geometric mean
If 3 numbers, say p1, p2 and p3, are
in geometric progression then the middle number p2 is the
geometric mean of p1 and p3.Since if p1 = a then p2 = ar and p3 = ar2 then:
Hence the geometric mean of x and y is:
Слайд 48Example 7
The 3rd term of a convergent GP is the
arithmetic mean of the 1st and 2nd terms. Find the
common ratio and, if the first term is 1, find the sum to infinity.We should consider the general term of a GP and the corresponding 1st, 2nd and 3rd term to find the common ratio. We can then use the formula for the sum to infinity.
Слайд 51Example 7 Solution
Now we know r and we are given
that a = 1 so we can use the formula
for the sum to infinity:
Слайд 52In summary
The general term of a GP is Tn =
arn – 1 where a is the 1st term, r
is the common ratio and n is the position number.The sum of the first n terms of a GP is given by:
For r < 0
For r > 0
Слайд 53In summary
A series may be the sum or difference of
an AP and a GP.
Therefore the sum to n
of the whole series will be the sum or difference of the sum to n of the AP and the sum to n of the GP.E.g. If the general term of a series is given by a1rn – 1 + a2 + (n – 1)d
then Sn = a1(1 – rn)/(1 – r) + n/2(a2 + l)
Слайд 54In summary
A series is convergent if its sum approaches a
specific value.
A series is divergent if its sum approaches (+/-)
infinity.All APs are divergent and so the sum to infinity is +/- infinity.
For GPs:
![Английская азбука
в картинках
A nt
А a
[ энт ] - муравей](/img/thumbs/5840ee849845b1fad1d13aaf1ddf3438-800x.jpg)
