节: 几何序列 | 13.6 几何序列

章节大纲

Introduction
::导言

A deposit of $200 is made on the 1st day of January, April, July, and October of every year in an account that pays 4.5% interest, compounded quarterly. The future balance can be modeled using a geometric series. An advanced factoring technique allows you to rewrite the sum of a finite geometric series in a compact formula. An infinite geometric series has two possibilities: It can sum to be a number, or the sum can continue to grow to infinity. When does an infinite geometric series sum to be just a number, and when does it sum to be infinity?
::每年1月1日、4月、7月和10月的1日存款200美元,存入一个支付4.5%利息的账户,每季交汇一次。未来余额可以使用几何序列来模拟。一种先进的计数技术可以使您在压缩公式中重写一个限定几何序列的总和。一个无限的几何序列有两种可能性:总和可以是一个数字,或者总和可以继续增长到无限。一个无限几何序列总和什么时候可以只是一个数字,总和什么时候可以无限?

Geometric Series
::几何序列

Recall the advanced factoring technique for the difference of two squares and, more generally, two terms of any power (5 in this case).
::回顾两个方形差异的先进保理技术,以及更一般而言,任何权力的两个条件(本案为5个)。

$\begin{aligned} a^{2} - b^{2} & = (a - b) (a + b) \\ a^{5} - b^{5} & = (a - b) (a^{4} + a^{3} b + a^{2} b^{2} + a b^{3} + b^{4}) \\ a^{n} - b^{n} & = (a - b) (a^{n - 1} + \dots + b^{n - 1}) \end{aligned}$

::a2-b2=(a-b)(a+b)(a+b)a5-b5=(a-b)(a4+a3b+a2b2+a3b3+b4)an-bn=(a-b)(an-1bn-1)

If the 1st term is 1, then $\begin{aligned} a = 1 \end{aligned}$ . If you replace $\begin{aligned} b \end{aligned}$ with the letter $\begin{aligned} r \end{aligned}$ , you end up with
::如果第一个任期为 1, 那么 a=1. 如果用字母 r 替换 b, 最终会使用

$\begin{aligned} 1 - r^{n} = (1 - r) (1 + r + r^{2} + \dots r^{n - 1}) . \end{aligned}$
You can divide both sides by $\begin{aligned} (1 - r) \end{aligned}$ because $\begin{aligned} r \neq 1 \end{aligned}$ .
::1-rn=(1-r)(1+r+r+r2_rn_1),您可以因 r_1 将两边除以(1-r)。

$\begin{aligned} 1 + r + r^{2} + \dots r^{n - 1} = \frac{1 - r^{n}}{1 - r} \end{aligned}$

::1+r+r2rn_1=1-rn1-r

The left side of this equation is a geometric series with starting term 1 and common ratio of $\begin{aligned} r \end{aligned}$ . Note that even though the ending exponent of $\begin{aligned} r \end{aligned}$ is $\begin{aligned} n - 1 \end{aligned}$ , there are a total of $\begin{aligned} n \end{aligned}$ terms on the left. To make the starting term not 1, just scale both sides of the equation by the 1st term you want, $\begin{aligned} a_{1} \end{aligned}$ .
::此方程的左侧是一个几何序列, 起始任期为 1 , 共同比率为 r 。请注意, 尽管 r 的结尾指数为 n- 1 , 但左侧总共有 n 条件。要使起始任期不是 1 , 只需将方程的两侧缩放到您想要的第一个条件 , a1 。

Sum of a Finite Geometric Series
::有限几何系列总和

$\begin{aligned} a_{1} + a_{1} r + a_{1} r^{2} + \dots a_{1} r^{n - 1} = a_{1} (\frac{1 - r^{n}}{1 - r}) \end{aligned}$

To sum an infinite geometric series, start by looking carefully at the previous formula for a finite geometric series. As the number of terms gets infinitely large, $\begin{aligned} (n \to \infty) \end{aligned}$ , one of two things will happen:
::与无限几何序列相加,首先仔细研究一个限定几何序列的先前公式。随着条件的数量变得无限之大,,两种情况之一将发生:

$\begin{aligned} a_{1} (\frac{1 - r^{n}}{1 - r}) \end{aligned}$

::a( 1 - rn1 - r)

Option 1: The term $\begin{aligned} r^{n} \end{aligned}$ will go to infinity or negative infinity. This will happen when $\begin{aligned} | r | \geq 1 \end{aligned}$ . When this happens, the sum of the infinite geometric series does not go to a specific number, and the series is said to be divergent.
::选项1: nn 术语将转到无穷或负无穷。当 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Option 2: The term $\begin{aligned} r^{n} \end{aligned}$ will go to zero. This will happen when $\begin{aligned} | r | < 1 \end{aligned}$ . When this happens, the sum of the infinite geometric series goes to a certain number, and the series is said to be convergent .
::选项2: nn 术语将变为零。当 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

One way to think about these options is think about what happens when you take $\begin{aligned} {0.9}^{100} \end{aligned}$ and $\begin{aligned} {1.1}^{100} \end{aligned}$ .
::一种思考这些选项的方法是思考一下,如果采取0.9100和1.1100,会发生什么。

$\begin{aligned} {0.9}^{100} & \approx 0.00002656 \\ {1.1}^{100} & \approx 13780 \end{aligned}$

As you can see, even numbers close to 1 either get very small quickly, or very large quickly.
::如你所见,甚至接近1的数要么很快变小,要么很快变大。

Sum of a Infinite Geometric Series
::无限几何系列总和

$\begin{aligned} \sum_{i = 1}^{\infty} a_{1} \cdot r^{i - 1} = a_{1} (\frac{1}{1 - r}) \end{aligned}$

Notice how this formula is the same as the finite version but with $\begin{aligned} r^{n} = 0 \end{aligned}$ , just as you reasoned.
::注意这个公式如何与限定版本相同, 但与 rn=0 相同, 正如你所解释的。

The following video introduces geometric series:
::以下录像介绍几何序列:

The following video explains how to determine the sum of an infinite geometric series, if the sum exists:
::下列录像解释如果存在一个无限几何序列,如何确定该数序列的总和:

Examples
::实例

Example 1
::例1

Compute the sum of the following infinite geometric series:
$\begin{aligned} 0.2 + 0.02 + 0.002 + 0.0002 + \dots \end{aligned}$
Solution:
::计算以下无限几何序列的总和: 0.2+0.02+0.002+0.0002+0.002+...解决方案:

You can tell just by looking at the sum that the infinite sum will be the repeating decimal $\begin{aligned} 0. \bar{2} \end{aligned}$ . You may recognize this as the fraction $\begin{aligned} \frac{2}{9} \end{aligned}$ , but if you don't, this is how you turn a repeating decimal into a fraction.
::您可以通过查看总和来分辨出无限总和将是重复的十进制小数点 0.2 。您可以确认这是第29项, 但如果您不这样做, 您就会将重复的十进制转换成一个小数点。

Let $\begin{aligned} x = 0. \bar{2} \end{aligned}$ .
::来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来,来

Then, $\begin{aligned} 10 x = 2. \bar{2} \end{aligned}$ .
::然后,10x2.2'。

Subtract the two equations and solve for $\begin{aligned} x \end{aligned}$ .
::减去两个方程式并解析 x。

$\begin{aligned} 10 x - x & = 2. \bar{2} - 0. \bar{2} \\ 9 x & = 2 \\ x & = \frac{2}{9} \end{aligned}$

::10x=2.2=0.2 9x=2x=29

Example 2
::例2

Why does an infinite series with $\begin{aligned} r = 1 \end{aligned}$ diverge ?
::为什么有r=1差异的无限序列?

Solution:
::解决方案 :

If $\begin{aligned} r = 1 \end{aligned}$ , this means that the common ratio between the terms in the is 1. This means that each number in the sequence is the same. When you add up an infinite number of any finite numbers (even fractions close to 0), you will always get infinity or negative infinity. The only exception is 0. This case is trivial because a geometric series with an initial value of 0 is simply the following series, which clearly sums to 0:
$\begin{aligned} 0 + 0 + 0 + 0 + \dots . \end{aligned}$
Example 3
::如果 r= 1, 这意味着在 1 中, 术语的通用比率为 r= 1 。这意味着序列中的每个数字是相同的。当您将任意的有限数字( 接近 0 的分数) 相加为无限数时, 你总是会得到无限数( 接近 0 的分数) 。唯一的例外是 0 。这个案例是微不足道的, 因为初始值为 0 的几何序列仅仅是以下序列, 明显等于 0: 0+0+0+0+0+. +.... 例如 3

What is the sum of the 1st eight terms in the following geometric series?
$\begin{aligned} 4 + 2 + 1 + \frac{1}{2} + \dots . \end{aligned}$
Solution:
::以下几何序列中第8个条件的总和是多少? 4+2+1+12+. 解决方案:

The 1st term is 4, and the common ratio is $\begin{aligned} \frac{1}{2} \end{aligned}$ .
$\begin{aligned} a_{1} (\frac{1 - r^{n}}{1 - r}) = 4 (\frac{1 - {(\frac{1}{2})}^{8}}{1 - \frac{1}{2}}) = 4 (\frac{\frac{255}{256}}{\frac{1}{2}}) = \frac{255}{32} \end{aligned}$
Example 4
::第一个学期为4个学期,共同比率为12个a1(1-1-rn1-r)=4(1-1-(12)81-12)=4(25525612)=25532例4。

Recall the question from the Introduction: W hen does an infinite geometric series sum to be just a number, and when does it sum to be infinity?
::回顾导言中的问题:无限几何序列总和何时只是数字,何时是无限?

Solution:
::解决方案 :

An infinite geometric series converges if and only if $\begin{aligned} | r | < 1 \end{aligned}$ . Infinite arithmetic series never converge .
::无限的几何序列如果而且只有在 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\"\\\\\\\\\"\\\\\\\\\\\\\\\\\"\\\\"\\\\\\\\"\\\\\\"\\\\\\\\\\\"\"\"\\"\"\"\"\"\"\"\"\"\\"\"\"\\"不无限的计算序列从不汇合。

Example 5
::例5

Compute the sum from Example 1 using the formula for the sum of an infinite geometric series, and confirm that the sum truly does converge.
::使用无限几何序列总和的公式,计算例1中的和,并确认该总和确实汇合。

Solution:
::解决方案 :

The 1st term of the sequence is $\begin{aligned} a_{1} = 0.2 \end{aligned}$ . The common ratio is 0.1. Since $\begin{aligned} | 0.1 | < 1 \end{aligned}$ , the series does converge.
$\begin{aligned} 0.2 (\frac{1}{1 - 0.1}) = \frac{0.2}{0.9} = \frac{2}{9} \end{aligned}$
Example 6
::序列的第一个条件为 a1=0.2. 。共同比率为 0.1。由于 =0. 1\\\ 1, 该序列的序列会合并。 0.2( 11- 0.1) =0. 20. 9 = 29例 6

Does the following geometric series converge or diverge? Does the sum go to positive or negative infinity?
$\begin{aligned} - 2 + 2 - 2 + 2 - 2 + \dots \end{aligned}$
Solution:
::以下几何序列相趋近还是相异?总和是正的还是负的无限?2+2-2+2-2+2+2+...解决方案:

The initial term is -2, and the common ratio is -1. Since the $\begin{aligned} | - 1 | \geq 1 \end{aligned}$ , the series is said to diverge. Even though the series diverges , it does not approach negative or positive infinity. When you look at the (the sums up to certain points), they alternate between two values:
$\begin{aligned} - 2, 0, - 2, 0, \dots . \end{aligned}$
This pattern does not go to a specific number. Just like a sine or cosine wave, it does not have a limit as it approaches infinity.
::初始术语是 - 2 , 共同比率是 - 1 。由于% 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = 1 = = 1 = 1 = 1 = 1 = 1 = 1 = 1 = = 1 = 1 = = 1 = 1 = 1 = = = = = 1 = 1 = 1 = = = 1 = = = 1 = 1, = = = 1 = = = = 1 = 1 = 1 = 1 = = = = = = = = = 1 = = = = = = = = = = = 1 = 1 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Example 7
::例7

You put $100 in a bank account at the end of every year for 10 years. The account earns 6% interest. How much do you have in total at the end of 10 years?
::您在10年内每年年底在银行帐户上存入100美元。该账户赚取6%的利息。在10年结束时,您总共有多少?

Solution:
::解决方案 :

The 1st deposit gains 9 years of interest: $\begin{aligned} 100 \cdot {1.06}^{9} \end{aligned}$
::第1期存款收益 9年利息:100 1.069

The 2nd deposit gains 8 years of interest: $\begin{aligned} 100 \cdot {1.06}^{8} \end{aligned}$ . This pattern continues, creating a geometric series. The last term receives no interest at all.
$\begin{aligned} 100 \cdot {1.06}^{9} + 100 \cdot {1.06}^{8} + \dots 100 \cdot 1.06 + 100 \end{aligned}$
Note that normally, geometric series are written in the opposite order so you can identify the starting term and the common ratio more easily.
$\begin{aligned} a_{1} = 100, r = 1.06 \end{aligned}$
The sum of the 10 years of deposits is:
$\begin{aligned} a_{1} (\frac{1 - r^{n}}{1 - r}) = 100 (\frac{1 - {1.06}^{10}}{1 - 1.06}) \approx $ 1, 318.08 \end{aligned}$

::第2期存款收益为8年利息:1001.068。这个模式继续,创建了一个几何序列。最后一个学期完全没有利息。 1001.069+1001.0681001001.0681001.006+100注。通常,几何序列以相反的顺序写,以便您能够更容易地识别起始期和共同比率。 a1=100,r=1.06。 10年存款的总和是:a1(1-rn1-r)=100(1-06101-1.06)1 318.08美元。

Summary
::摘要

To converge means the sum approaches a specific number.
::趋同意味着求和法采用一个具体的数字。

To diverge means the sum does not converge, and so usually goes to positive or negative infinity. It could also mean that the series oscillates infinitely.
::差异意味着总和不会趋同,因此通常会变成正或负的无限。它还可能意味着序列会无穷地振动。

A partial sum of an infinite sum is the sum of all the terms up to a certain point. Considering partial sums can be useful when analyzing infinite sums.
::无限总和的一部分是直到某一点的所有条件的总和。在分析无限总和时,考虑部分总和是有用的。

Notation for the sum of a geometric series: $\begin{aligned} a_{1} + a_{1} r + a_{1} r^{2} + \dots a_{1} r^{n - 1} = a_{1} (\frac{1 - r^{n}}{1 - r}) \end{aligned}$ .
::a1+a1r+a1r2_a1rn_1=a1(1-rn1-r)。

Review
::回顾

Find the sum of the 1st 15 terms for each geometric sequence below.
::为以下每一几何序列查找第15个条件的总和。

1. $\begin{aligned} 5, 10, 20, \dots \end{aligned}$

2. $\begin{aligned} 2, 8, 32, \dots \end{aligned}$

3. $\begin{aligned} 5, \frac{5}{2}, \frac{5}{4}, \dots \end{aligned}$

4. $\begin{aligned} 12, 4, \frac{4}{3}, \dots \end{aligned}$

5. $\begin{aligned} \frac{1}{3}, 1, 3, \dots \end{aligned}$

For each infinite geometric series, identify whether the series is convergent or divergent. If convergent, find the number where the sum converges.
::对于每个无限的几何序列,请标明该序列是集合还是相异。如果是集合,请找到总和相交的编号。

6. $\begin{aligned} 5 + 10 + 20 + \dots \end{aligned}$

7. $\begin{aligned} 2 + 8 + 32 + \dots \end{aligned}$

8. $\begin{aligned} 5 + \frac{5}{2} + \frac{5}{4} + \dots \end{aligned}$

9. $\begin{aligned} 12 + 4 + \frac{4}{3} + \dots \end{aligned}$

10. $\begin{aligned} \frac{1}{3} + 1 + 3 + \dots \end{aligned}$

11. $\begin{aligned} 6 + 2 + \frac{2}{3} + \dots \end{aligned}$

12. You put $5,000 in a bank account at the end of every year for 30 years. The account earns 2% interest. How much do you have in total at the end of 30 years?
::12. 30年来,每年年底在银行帐户上放5 000美元,该帐户赚取2%的利息。30年结束时,你总共有多少利息?

13. You put $300 in a bank account at the end of every year for 15 years. The account earns 4% interest. How much do you have in total at the end of 10 years?
::13. 15年来,每年年底在银行帐户上存入300美元,该帐户赚取4%的利息。 10年结束时,你总共有多少利息?

14. You put $10,000 in a bank account at the end of every year for 12 years. The account earns 3.5% interest. How much do you have in total at the end of 12 years?
::14. 12年中,每年年底在银行帐户上存入1万美元,每年赚取3.5%的利息。12年结束时,你总共有多少利息?

15. Why don’t infinite arithmetic series converge?
::15. 为什么无限的算术系列不汇合?

Review (Answers )
::回顾(答复)

Please see the Appendix.
::请参看附录。

章节大纲

Introduction ::导言

Geometric Series ::几何序列

Sum of a Finite Geometric Series ::有限几何系列总和

Sum of a Infinite Geometric Series ::无限几何系列总和

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers ) ::回顾(答复)

Introduction
::导言

Geometric Series
::几何序列

Sum of a Finite Geometric Series
::有限几何系列总和

Sum of a Infinite Geometric Series
::无限几何系列总和

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers )
::回顾(答复)