几何序列

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An advanced factoring technique allows you to rewrite the sum of a finite geometric series in a compact formula. An infinite geometric series is more difficult because sometimes it sums to be a number and sometimes the sum keeps on growing to infinity. When does an infinite geometric series sum to be just a number and when does it sum to be infinity?
::高级乘数技术允许您重写压缩公式中限定几何序列的总和。 无限几何序列更为困难, 因为有时它总算为一个数字, 有时它又继续增长到无限。 当一个无限几何序列总和只是一个数字, 什么时候它总和为无限?

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Geometric Series
::几何序列

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A geometric series is a sum of numbers whose consecutive terms form a geometric . Recall the advanced factoring technique for the difference of two squares and, more generally, two terms of any power (5 in this case).
::几何序列是数字的总和,其连续条件构成几何。 回顾两个平方之间的差数,更一般地回顾任何权力的两个条件(在本案中为5个)。

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$$\begin{align*}a^2-b^2&=(a-b)(a+b) \\ a^5-b^5&=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4) \\ a^n-b^n&=(a-b)(a^{n-1}+ \cdots +b^{n-1})\end{align*}$$
::a2-b2=(a-b)(a+b)(a+b)a5-b5=(a-b)(a4+a3b+a2b2+a3b3+b4)an-bn=(a-b)(an-1bn-1)

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If the first term is one then $\begin{align*}a=1\end{align*}$ . If you replace $\begin{align*}b\end{align*}$  with the letter $\begin{align*}r\end{align*}$ , you end up with:
::如果第一个任期为一任期,则 a=1. 如果用字母 r 替换b, 以下列方式结束:

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$\begin{align*}1-r^n=(1-r)(1+r+r^2+ \cdots r^{n-1})\end{align*}$
::1-rn=(1-r)(1+r+r+r2__rn-1)

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You can divide both sides by $\begin{align*}(1-r)\end{align*}$   because $\begin{align*}r \ne 1\end{align*}$ .
::你可以将两边除以(1-r),因为 r% 1 。

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$\begin{align*}1+r+r^2+\cdots r^{n-1}=\frac{1-r^n}{1-r} \end{align*}$
::1+r+r2rn_1=1-rn1-r

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The left side of this equation is a geometric series with starting term 1 and common ratio of $\begin{align*}r\end{align*}$ . Note that even though the ending exponent of  $\begin{align*}r\end{align*}$ is $\begin{align*}n-1\end{align*}$ , there are a total of  $\begin{align*}n\end{align*}$ terms on the left. To make the starting term not one, just scale both sides of the equation by the first term you want, $\begin{align*}a_1\end{align*}$ .
::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 。 要使起始任期不是一, 只需在第一个任期前将方程的两边比例都标定为 a1 。

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$\begin{align*}a_1+a_1r+a_1r^2+\cdots a_1r^{n-1}=a_1 \left(\frac{1-r^n}{1-r}\right)\end{align*}$
::a1+a1r+a1r2+a1r2}a1rn_1=a1(1-rn1-r)

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This is the sum of a finite geometric series .
::这是有限的几何序列的总和。

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To sum an infinite geometric series , you should start by looking carefully at the previous formula for a finite geometric series. As the number of terms get infinitely large  $\begin{align*}(n\rightarrow \infty)\end{align*}$ one of two things will happen.
::与无限几何序列相加,您应该从仔细查看一个限定几何序列的先前公式开始。随着条件数量变得无限大,两种条件中的一种就会发生。

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$\begin{align*}a_1 \left(\frac{1-r^n}{1-r}\right)\end{align*}$
::a( 1 - rn1 - r)

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Option 1:  The term  $\begin{align*}r^n\end{align*}$ will go to infinity or negative infinity. This will happen when $\begin{align*}\left |r \right \vert \ge 1\end{align*}$ . 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 术语将进入无穷或负无穷。 当 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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Option 2:  The term  $\begin{align*}r^n\end{align*}$ will go to zero. This will happen when $\begin{align*}\left |r \right \vert < 1\end{align*}$ . 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 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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One way to think about these options is think about what happens when you take  $\begin{align*}0.9^{100}\end{align*}$ and $\begin{align*}1.1^{100}\end{align*}$ .
::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。

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

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As you can see, even numbers close to one either get very small quickly or very large quickly.
::如你所见,即使数字接近一个数字, 也可能很快变小或变大。

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The formula for calculating the sum of an infinite geometric series that converges is:
::计算相交的无限几何序列之和的公式是:

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$\begin{align*}\sum \limits_{i=1}^{\infty} a_1 \cdot r^{i-1}=a_1 \left(\frac{1}{1-r}\right)\end{align*}$
::i=1a1-i-1=1a1-1=a1(11-r)

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Notice how this formula is the same as the finite version but with $\begin{align*}r^n=0\end{align*}$ , 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 。

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A partial sum of an infinite sum is the sum of all the terms up to a certain point. Considering 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,考虑是有用的。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked when a infinite geometric series sum to just a number. An infinite geometric series converges if and only if $\begin{align*}\left |r\right \vert <1\end{align*}$ . Infinite arithmetic series never converge .
::早些时候,有人问您一个无限的几何序列和一个数字是什么时候。一个无限的几何序列如果和只有如果和只有如果\\\\\\\\\\\\\\\\\\\无穷的计算序列之间会一致的话,那么无限的几何序列和一个数字是什么时候。一个无限的几何序列如果和只有如果和只有如果\\\\\\\\\\\\\\\\\\\无穷的计算序列永远不会一致的话。

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Example 2
::例2

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Compute the sum of the following infinite geometric series two ways, without using the infinite summation formula and using the infinite summation formula.
::以两种方式计算以下无限几何序列的和,不使用无限相加公式,也不使用无限相加公式。

$\begin{align*}0.2+0.02+0.002+0.0002+\cdots \end{align*}$

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Without using the summation formula:
::不使用总和公式:

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You can tell just by looking at the sum that the infinite sum will be the repeating decimal  $\begin{align*}0.\overline{2} \end{align*}$ . You may recognize this as the fraction $\begin{align*}\frac{2}{9}\end{align*}$ , 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小数小数点 0 。 您可以确认这是第29项, 但如果您不这样做, 您就会将重复的小数点转换成一个小数点 。

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Let $\begin{align*}x=0.\overline{2}\end{align*}$
::让我们x=0。 $2

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Then $\begin{align*}10x=2.\overline{2}\end{align*}$
::然后10x2分 10x2分

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

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$$\begin{align*}10x-x&=2.\overline{2}-0.\overline{2} \\ 9x&=2 \\ x&=\frac{2}{9}\end{align*}$$
::10x2 2x2 2x0 29x2x29

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With using the summation formula:
::使用总和公式:

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The first term of the sequence is $\begin{align*}a_1=0.2\end{align*}$ . The common ratio is 0.1. Since $\begin{align*}\left |0.1 \right \vert <1\end{align*}$ , the series does converge.
::序列的第一个条件为 a1=0.2. 。 常见比率为 0.1。 自 0. 1\\\ 1 以来, 该序列的序列的确会趋同 。

$\begin{align*}0.2 \left(\frac{1}{1-0.1}\right)=\frac{0.2}{0.9}=\frac{2}{9}\end{align*}$

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Example 3
::例3

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

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If  $\begin{align*}r=1\end{align*}$ this means that the common ratio between the terms in the sequence 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 zero) 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:
::如果 r= 1 表示序列中术语的共比为 1 。 这意味着序列中的每个数字都是相同的。 当您将任何有限数字(甚至接近于零的分数)加在一起时, 你总是会得到无限数的无穷或负无穷。 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, 其数值明显等于 0 :

$\begin{align*}0+0+0+0+\cdots\end{align*}$

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Example 4
::例4

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What is the sum of the first 8 terms in the following geometric series?
::以下几何序列中前8个术语的总和是多少?

$\begin{align*}4+2+1+\frac{1}{2}+\cdots\end{align*}$

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The first term is 4 and the common ratio is $\begin{align*}\frac{1}{2}\end{align*}$ .
::第一个学期为4年,共同比率为12年。

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$\begin{align*}SUM=a_1\left(\frac{1-r^n}{1-r}\right)=4 \left(\frac{1- \left(\frac{1}{2}\right)^8}{1- \frac{1}{2}}\right)=4 \left(\frac{\frac{255}{256}}{\frac{1}{2}}\right)=\frac{255}{32}\end{align*}$
::SUM=a1(1-rn1-r)=4(1-(12)81-12)=4(25525612)=25532

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Example 5
::例5

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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 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。该帐户赚取6%的利息。在10年结束时,你总共有多少?

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The first deposit gains 9 years of interest: $\begin{align*}100 \cdot 1.06^9\end{align*}$
::首期存款收益 9年利息: 100 1.069

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The second deposit gains 8 years of interest: $\begin{align*}100 \cdot 1.06^8\end{align*}$ . This pattern continues, creating a geometric series. The last term receives no interest at all.
::第二个存款收益是8年的利息:1001.068。这个模式继续,创造了几何序列。最后一个任期完全没有利息。

$\begin{align*}100 \cdot 1.06^9+100 \cdot 1.06^8+\cdots 100 \cdot 1.06+100\end{align*}$

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Note that normally geometric series are written in the opposite order so you can identify the starting term and the common ratio more easily.
::请注意,通常几何序列以相反的顺序写成,这样可以更容易地识别起始期和共同比率。

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$\begin{align*}a_1=100,r=1.06\end{align*}$
::a1=100,r=1.06

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The sum of the 10 years of deposits is:
::10年的存款总和如下:

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$\begin{align*}a_1\left(\frac{1-r^n}{1-r}\right)=100 \left(\frac{1-1.06^{10}}{1-1.06}\right)\approx \$1318.08\end{align*}$
::a1(1-rn1-r)=100(1-1-06101-1.06) =1318.08美元

Summary

播放段落 A geometric series is a sum of numbers whose consecutive terms form a geometric sequence, and it can be finite or infinite. ::几何序列是数字的总和,其连续条件构成几何序列,可以是有限的,也可以是无限的。 播放段落 The sum of a finite geometric series can be calculated using the formula: $\begin{align*}a_1+a_1r+a_1r^2+\cdots a_1r^{n-1}=a_1 \left(\frac{1-r^n}{1-r}\right),\end{align*}$  where $\begin{align*}a_1\end{align*}$  is the first term, $\begin{align*}r\end{align*}$  is the common ratio, and $\begin{align*}n\end{align*}$  is the number of terms. ::a1+a1r+a1r2a1rn_1=a1(1-rn1-r),a1是第一个学期,r是共同比率,n是术语数。 播放段落 An infinite geometric series can be either convergent or divergent, depending on the value of the common ratio $\begin{align*}r.\end{align*}$   ::一个无限的几何序列可以是趋同的,也可以是不同的,取决于共同比率r的值。 播放段落 If $\begin{align*}|r| \ge 1,\end{align*}$  the infinite geometric series is divergent, and its sum does not go to a specific number. ::如果 \\\\\\\\\\\\\\\\ i,无限几何序列是不同的, 其总和没有去到一个特定的数字 。 播放段落 If |r| < 1, the infinite geometric series is convergent, and its sum can be calculated using the formula: $\begin{align*}\sum \limits_{i=1}^{\infty} a_1 \cdot r^{i-1}=a_1 \left(\frac{1}{1-r}\right)\end{align*}$   ::如果 \ r < 1, 无限几何序列是聚合的, 其总和可以使用公式计算 :\ i= 1a111- 1=a1( 11- r) 播放段落 A partial sum of an infinite sum is the sum of all the terms up to a certain point. ::无限数额的部分总和是直至某一点的所有条件的总和。

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Review
::回顾

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

1. $\begin{align*}5,10,20,\ldots\end{align*}$

2. $\begin{align*}2,8,32,\ldots\end{align*}$

3. $\begin{align*}5,\frac{5}{2},\frac{5}{4},\ldots\end{align*}$

4. $\begin{align*}12,4,\frac{4}{3},\ldots\end{align*}$

5. $\begin{align*}\frac{1}{3},1,3,\ldots\end{align*}$

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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. $\begin{align*}5+10+20+\cdots\end{align*}$

7. $\begin{align*}2+8+32+\cdots\end{align*}$

8. $\begin{align*}5+\frac{5}{2}+\frac{5}{4}+\cdots\end{align*}$

9. $\begin{align*}12+4+\frac{4}{3}+\cdots\end{align*}$

10. $\begin{align*}\frac{1}{3}+1+3+\cdots\end{align*}$

11. $\begin{align*}6+2+\frac{2}{3}+\cdots\end{align*}$

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12. You put $5000 in a bank account at the end of every year for 30 years. The account earns 2% interest. How much do you have total at the end of 30 years?
::12. 30年来,你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?

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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 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?

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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 total at the end of 12 years?
::14. 12年来,每年年底在银行帐户上存入1万美元,该帐户赚取3.5%的利息。 12年结束时,你总共有多少利息?

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

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Review (Answers)
::回顾(答复)

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Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。