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  • Better Living Through Mathematics! with Professor Polly Ethylene

    通過數學讓生活更美好!與Polly Ethylene教授一起分享。

  • Infinite series is one of the most fascinating subjects in mathematics.

    無限系列是數學中最有魅力的學科之一。

  • But before we talk about infinite series, I would like to ask a question about infinity.

    但在我們談論無限系列之前,我想問一個關於無限的問題。

  • What do you get when you add an infinite number of things?

    當你把無限多的東西加在一起,你會得到什麼?

  • A.V. Geekman, do you have an answer up that sleeve of yours?

    A. V. Geekman, 你的袖子裡有答案嗎?

  • Well, the more things you add up, the bigger the answer gets.

    嗯,你加的東西越多,答案就越大。

  • So if you were to add up an infinite number of things

    所以,如果你要把無限多的東西加起來。

  • you would always get an infinitely large answer.

    你總會得到一個無限大的答案。

  • Are you sure about that A.V.?

    你確定那個A. V. ?

  • Well, no matter what size the things are

    好吧,不管東西是什麼大小

  • an infinite number of them will always be infinity.

    無限的數量將永遠是無限的。

  • For example, an infinite number of "ones" is infinity times one, or infinity.

    例如,無限個"一"是一的無限次,或無限大。

  • An infinite number of one-halves is infinity times one-half.

    無數個一角就是無限乘一角。

  • One half of infinity is still infinity.

    無限的一半還是無限。

  • Even an infinite number of one-billionths is one-billionth of infinity

    即使是無限的十億分之一,也是無限的十億分之一。

  • which is still infinity!

    這仍然是無限的!

  • Ah ha!

    啊哈!

  • That's extremely perceptive of you A.V.

    你真是太有洞察力了 A. V.

  • It is true that an infinite number of anything, no matter how small, is still infinitely large.

    誠然,無限多的任何東西,無論多小,還是無限大。

  • So, another question to you.

    所以,另一個問題要問你。

  • Is it possible to add an infinite number of things and get a finite number?

    是否可以把無限多的東西加起來,得到一個有限的數?

  • Well, based upon our previous discussion, I would say not.

    好吧,根據我們之前的討論,我會說不是。

  • Well then, you would be wrong!

    那麼,你就錯了!

  • Oh really?

    哦,真的嗎?

  • Now A.V., as you pointed out

    現在,A. V. ,正如你所言

  • if you add an infinite number of anything, no matter how small

    如果你把無限的東西加進去,無論多小的東西都是如此

  • the sum is still infinite.

    總和仍然是無限的。

  • But what if the things you add get progressively smaller?

    但如果你添加的東西逐漸變小怎麼辦?

  • I ... don't know.

    我......不知道。

  • All right then!

    好吧!

  • Let's add an infinite number of things, where each thing is one-half of the previous thing.

    讓我們'加無限多的東西,其中每個東西是前一個東西的二分之一。

  • Let's say, for example, that you walk halfway to the wall.

    比方說,你走到一半的牆邊。

  • Then you walk half of the remaining distance

    然後你再走一半的路程

  • then half of that distance

    半程

  • and half of that distance, and so on.

    以及該距離的一半,以此類推。

  • No matter how many times you keep doing that, you will still never quite reach the wall.

    無論你繼續做多少次,你還是永遠不會完全到達牆邊。

  • So it IS possible to add an infinite number of things and get a finite number!

    所以,把無限的東西加起來,得到的是有限的數,這是有可能的!

  • Now you're cooking with gas!

    現在你用煤氣做飯了!

  • I'll give you one more example.

    我'再舉一個例子。

  • Let's take the fraction nine-tenths

    讓我們以分數的十分之九為例。

  • which you can write as the decimal number zero point nine.

    可以寫成十進制數零點九。

  • Now let's add nine-hundredths

    現在讓我們加上九百分之一

  • and nine-thousandths

    和千分之九

  • and so on.

    諸如此類。

  • As you can see, we can keep doing this as long as you like

    如你所見,只要你喜歡,我們可以一直這樣做下去

  • but you will never get a number bigger than one.

    但你永遠不會得到一個大於1的數字。

  • We say the "limit" of this series of additions is one.

    我們說"極限"這一系列的加法是一。

  • So, let's talk about what a series is.

    那麼,讓我們來談談什麼是系列。

  • A series is just a list of things added together.

    一個系列就是把一系列的東西加在一起。

  • These things can be numbers

    這些東西可以是數字

  • or they can be expressions or formulas which create numbers.

    或者它們可以是創建數字的表達式或公式。

  • Let's call these cute little things that are added together "terms".

    讓我們把這些可愛的小東西加在一起叫做"條款"。

  • So a series is just a sum of terms.

    所以一個系列只是一個項的總和。

  • Now, if I wanted you to add a bunch of terms for me

    現在,如果我想讓你給我加一堆術語的話。

  • I might write down all of the terms in a list.

    我可能會把所有的術語都寫在一個列表裡。

  • That list of terms is called a "sequence"

    這個術語列表稱為"序列&quot。

  • and when you add all the terms in a sequence together, it is called a "series".

    而當你把一個序列中的所有項加在一起時,就叫做"序列"。

  • Now, what if the sequence has a very large number of terms?

    現在,如果這個序列有非常多的項,怎麼辦?

  • In fact, what if the sequence is infinite?

    其實,如果序列是無限的呢?

  • It might be easier to come up with a formula for creating the terms

    也許更容易得出一個公式來創建術語。

  • instead of having to list each one.

    而不必逐一列出。

  • Now wouldn't that be a good idea?

    現在,這不是一個好主意嗎?

  • Wonderful!

    妙極了!

  • Now, being the good perceptive professor that you are

    現在,作為一個好的洞察力教授,你是誰?

  • you might have noticed there was a pattern to the terms I wrote on your list.

    你可能已經注意到了我在你的清單上寫的術語有一個模式。

  • If we number the terms one, two, three, etc. then each term is just twice that number.

    如果我們對一、二、三等項進行編號,那麼每個項只是這個數字的兩倍。

  • So we can specify this sequence by writing a formula.

    所以我們可以通過寫一個公式來指定這個序列。

  • This formula says that each term

    這個公式說,每項

  • which we will call "a" numbered with a little subscript "n"

    我們稱之為"a",用一個小標"n&quot編號。

  • which tells us which term in the sequence it is

    它告訴我們它是序列中的哪項

  • is just two times n.

    只是n的2倍。

  • So for instance, the twentieth term, "a" sub twenty

    所以,比如第二十項,"a"次二十項。

  • is just two times twenty, or forty.

    就是二乘二十,也就是四十。

  • Now isn't that easier than listing every term?

    現在不是'比把每個術語都列出來更容易嗎?

  • Of course, we could make the formula for creating terms as complicated as we like

    當然,我們可以把創造術語的公式弄得越複雜越好。

  • such as

    諸如

  • or

  • or even something really complicated!

    甚至是很複雜的東西!

  • But the point is, if we can come up with a formula for creating the terms in the sequence

    但問題是,如果我們能想出一個公式來創建序列中的術語。

  • we can write the sequence in a very compact form and save a lot of paper.

    我們可以把序列寫成非常緊湊的形式,節省大量的紙張。

  • Thank you for showing the class how to write an infinite sequence, Professor Ethylene

    謝謝你向同學們展示瞭如何寫出一個無限序列,乙烯教授

  • but weren't you going to explain to the class about infinite series?

    但你不是要向全班同學解釋無限系列嗎?

  • Well thank you Professor Von Schmohawk for reminding me of that fact.

    好吧,謝謝你馮-施莫霍克教授提醒我這個事實。

  • As I mentioned, a "series" is just the terms of a "sequence" added together.

    正如我所提到的,"系列"只是一個"序列"的條款加在一起。

  • Now, there is a nice way to write a series

    現在,有一個很好的方法來寫一個系列。

  • using what we mathematicians call "summation notation".

    使用我們數學家所說的"求和符號"。

  • Here is the summation notation for the first five terms in the sequence

    以下是序列中前五項的求和符號

  • which I wrote down for the good Professor Von Schmohawk.

    這是我為好心的馮-施莫霍克教授寫下的。

  • The "summation symbol" is the capital Greek letter "sigma".

    符號"summation"是大寫的希臘字母"sigma"。

  • It indicates that the term to the right

    它表示右邊的術語

  • is added over and over again

    一遍遍地加

  • each time, using a different value of n.

    每次使用不同的n值。

  • In this case, n starts at one.

    在這種情況下,n從1開始。

  • This value of n is then used to calculate the first term of the series.

    然後用這個n的值來計算系列的第一項。

  • Since n is one 2n is equal to two times one

    由於n是1 2n等於2乘以1

  • or two.

    或兩個。

  • So two is the first term in the series.

    所以二是系列中的第一項。

  • Then we increase n by one, and do it again.

    然後我們把n增加一,再做一次。

  • This calculates the second term of the series.

    這可以計算系列的第二項。

  • We keep doing this until n finally reaches the value at the top of the summation symbol.

    我們一直這樣做,直到n最終達到求和符號頂端的值。

  • So this summation notation is another way of writing these five terms added together.

    所以這個求和符號是這五個術語相加的另一種寫法。

  • The sum of the terms is thirty

    條款的總和是30

  • so this finite series is equal to thirty.

    所以這個有限系列等於三十。

  • What if instead of stopping when n equals five, we went on forever?

    如果我們不在n等於5的時候停下來,而是永遠地走下去呢?

  • In this case, instead of the five at the top, we would put a little infinity sign.

    在這種情況下,我們會在頂部放一個小的無限符號,而不是5。

  • This would then be an "infinite" series.

    那麼這將是一個"無限"系列。

  • Now we would keep adding terms forever.

    現在,我們將永遠增加條款。

  • In this series the terms get bigger and bigger, so the sum is obviously infinite.

    在這個系列中,條件越來越大,所以總和顯然是無限的。

  • But even if the terms were all the same number, the sum would still be infinite.

    但即使條件都是相同的數字,總和仍然是無限的。

  • Take for example an infinite series where all the terms are the number one

    以一個無限數列為例,其中所有的項都是第1項。

  • or one-half.

    或一半。

  • In fact, adding any number that's not zero an infinite number of times gives you infinity.

    事實上,將任何一個不為零的數字無限次相加,都會得到無窮大。

  • So as long as the terms grow or stay the same

    所以,只要條件增長或保持不變

  • an infinite number of them will always sum up to infinity.

    無限的數量,其總和總是無限的。

  • But what would happen if each term was smaller than the previous term?

    但如果每一屆都比上一屆小,會怎樣呢?

  • Let's take an infinite series of the terms one over two to the nth power.

    讓我們'把一個無限系列的項一過二到n次方。

  • In this series the first term is one over two to the first power, or one-half.

    在這個系列中,第一項是一過二到一的冪,或二分之一。

  • The second term is one over two squared, or one-fourth.

    第二項是一過二平方,即四分之一。

  • The third term is one over two cubed, or one-eighth,

    第三項是一超二立方,即八分之一。

  • and so on.

    諸如此類。

  • Let's draw a picture of what happens when we add the terms in this series.

    讓我們'畫出一幅畫,當我們在這個系列中添加術語時會發生什麼。

  • Start by drawing a square with a length and height of one

    先畫一個長寬高各為1的正方形,然後再畫一個

  • so that the square has an area of one.

    所以,正方形的面積為1。

  • Now, the first term of our series is one-half

    現在,我們系列的第一項是二分之一。

  • so draw a rectangle with an area of half of the square

    畫一個面積為正方形一半的長方形

  • and place it in the square.

    並將其放置在廣場上。

  • Now the second term of the series is one-fourth.

    現在系列的第二項是四分之一。

  • Let's draw a square with an area of half of the rectangle

    讓我們來畫一個面積為長方形一半的正方形吧

  • and place it in the square.

    並將其放置在廣場上。

  • The third term of our series is one-eighth

    我們系列的第三項是八分之一。

  • so let's draw a rectangle with an area of half of the previous square

    所以讓我們畫一個面積為前一個正方形一半的長方形吧

  • and place it in the square.

    並將其放置在廣場上。

  • This process can be repeated forever without overflowing the square.

    這個過程可以永遠重複,不會溢出廣場。

  • As the little squares and rectangles continue to add up

    隨著小方塊和小長方形的不斷增加。

  • their total area becomes closer and closer to the area of the big square.

    他們的總面積越來越接近大廣場的面積。

  • The combined area of the terms gets closer and closer to one.

    條款的組合面積越來越接近於一個。

  • If you could add an infinite number of these terms, the total area would be exactly one.

    如果你能把這些條款無限地加起來,總面積正好是一個。

  • So we say that this series "converges" to one.

    所以我們說這個數列"收斂"為一。

  • In other words, this series is "convergent".

    換句話說,這個系列是"收斂"。

  • Convergent series are very useful.

    收斂系列是非常有用的。

  • Some numbers like pi can only be calculated by using convergent infinite series.

    有些數如pi只能用收斂無限數列來計算。

  • Here are the first few terms of an infinite series which can be used to calculate pi.

    這裡是無限數列的前幾項,可以用來計算pi。

  • Of course we can't actually add an infinite number of terms

    當然,我們實際上不可能增加無限多的條款'。

  • unless we had an infinite amount of time.

    除非我們有無限的時間。

  • However, we can make our answer as accurate as we like by simply adding enough terms.

    不過,我們只要增加足夠的條件,就可以使我們的答案儘可能的準確。

  • But will all series converge as long as each term is smaller than the previous one?

    但是,只要每個項比前一個項小,所有的系列都會收斂嗎?

  • Well, let's try a series with the terms one over n.

    好吧,讓我們試試一系列的術語一過n。

  • Now, the first term in this series is one divided by one

    現在,這個系列的第一項是1除以1

  • or one.

    或一。

  • The second term is one divided by two, or one-half.

    第二項是一除二,即二分之一。

  • The third term is one-third, and so on.

    第三屆為三分之一,以此類推。

  • Each term is smaller than the previous term.

    每一屆都比上一屆小。

  • But it turns out that this series does NOT converge.

    但事實證明,這個系列並不收斂。

  • Even though the terms get smaller and smaller, they will still add up to infinity.

    即使條款越來越小,但加起來還是會變成無窮大。

  • We say that this series "diverges".

    我們說,這個系列"分歧"。

  • Perhaps it seems strange that some series with decreasing terms converge to a number

    也許看起來很奇怪,一些具有遞減項的數列收斂到了一個數字上

  • while other series with decreasing terms diverge to infinity.

    而其他具有遞減項的數列則向無窮大發散。

  • It is not always obvious which series will converge or diverge.

    哪些系列會收斂或發散並不總是很明顯。

  • Let's take a closer look at this series to see why it never converges.

    讓我們仔細看看這個系列,看看它為什麼從不收斂。

  • Let's write down the first few terms of this infinite series.

    讓我們寫下這個無限系列的前幾項。

  • Now, let's make little stacks equal in height to each term in the series.

    現在,讓我們把系列中的每個項做成高度相等的小堆棧。

  • Notice that the first term of the series is equal to one-half plus one-half.

    注意,系列的第一項等於二分之一加二分之一。

  • The second term in the series is also one-half.

    系列中的第二任期也是二分之一。

  • Now notice that the next two terms, one-third and one-fourth

    現在請注意,接下來的兩個詞,三分之一和四分之一

  • are each at least as big as one-fourth.

    各自至少有四分之一那麼大。

  • So, if we add them together, their sum will be at least as big as one-fourth plus one-fourth

    所以,如果我們把它們加在一起,它們的總和至少有四分之一加四分之一那麼大。

  • or one-half.

    或一半。

  • Now the next four terms, one-fifth, one-sixth, one-seventh, and one-eighth

    現在,接下來的四屆,五分之一,六分之一,七分之一,八分之一。

  • are each at least as big as one-eighth.

    都至少有八分之一那麼大。

  • So, if we add them together, their sum will be at least as big as four times one-eighth

    所以,如果我們把它們加在一起,它們的總和至少有四倍八分之一那麼大。

  • or, once again, one-half.

    或者,再一次,二分之一。

  • Likewise, the next eight terms, one-ninth, one-tenth, one-eleventh, one-twelfth, one-thirteenth

    同樣,接下來的八屆,第九屆、第十屆、第十一屆、第十二屆、第十三屆。

  • one-fourteenth, one-fifteenth, and one-sixteenth

    十四分之一、十五分之一和十六分之一

  • are each at least as big as one-sixteenth.

    各自至少有十六分之一那麼大。

  • So when we add them together, their sum will be bigger than eight times one-sixteenth

    所以,當我們把它們加在一起的時候,它們的總和就會大於八倍十六分之一。

  • or, once again, one-half.

    或者,再一次,二分之一。

  • Likewise, the sum of the next sixteen terms is bigger than one-half

    同樣,接下來的十六個條款的總和也大於二分之一

  • and the sum of the next thirty-two terms is bigger than one-half

    而接下來的三十二項的總和大於二分之一。

  • and so on.

    諸如此類。

  • We can keep going on forever, grouping the terms into sums which equal more than one-half.

    我們可以永遠繼續下去,把這些條款歸為等於二分之一以上的總和。

  • So the sum of this infinite series is at least as big as

    所以,這個無限系列的總和至少有以下這麼大。

  • the sum of an infinite number of one-halves

    二分之一之和

  • which is of course, infinite.

    當然,這是無限的。

  • This particular series is called a "harmonic series"

    這個特殊的系列被稱為"諧波系列&quot。

  • because its terms are similar to the harmonics of a musical note.

    因為它的術語與音符的諧音相似。

  • Oh, I diverge!

    哦,我不同意!

  • Although the harmonic series is interesting

    雖然諧波系列很有趣

  • it is not very useful because its sum never converges.

    它的用處不大,因為它的和永遠不會收斂。

  • Are there any questions?

    有什麼問題嗎?

  • Hulk Moosemasher, what is your question?

    綠巨人摩斯馬舍,你的問題是什麼?

  • Professor Ethylene, are infinite series useful?

    乙烯教授,無限系列有用嗎?

  • Why yes Hulk, infinite series are very useful!

    為什麼是綠巨人,無限系列非常有用!

  • There are many things which can only be calculated by using infinite series.

    有很多東西只能用無限系列來計算。

  • For example, the ratio of the circumference of a circle to its diameter is pi.

    例如,圓的周長與直徑的比值是pi。

  • For many centuries, people measured circles but could never determine exactly

    許多世紀以來,人們測量圓的尺寸,但始終無法準確地確定。

  • what this ratio was to an accuracy of more than a few decimal places.

    這個比例是多少,精確到小數點後幾位。

  • But with the help of an infinite series

    但是,在無限系列的幫助下

  • we can determine pi to any degree of accuracy we like.

    我們可以確定圓周率的任何程度的準確性,我們喜歡。

  • The more terms in the series we add, the more accurate our answer gets.

    我們在系列中添加的術語越多,我們的答案就越準確。

  • Infinite series are also used to calculate trigonometric functions such as sine and cosine

    無限級數也用於計算三角函數,如正弦和餘弦。

  • which are very useful in determining the angles and lengths of triangles

    在確定三角形的角和長時非常有用。

  • as well as exponential functions, logarithms

    以及指數函數,對數函數

  • and many other mathematical functions which are used in engineering, science, and math.

    以及許多其他數學函數,這些函數在工程、科學和數學中都有應用。

  • I hope this answered your questions.

    我希望這能回答你的問題。

  • And remember, just like a ninety degree angle

    記住,就像九十度角一樣。

  • I'm always RIGHT!

    我永遠是對的!

Better Living Through Mathematics! with Professor Polly Ethylene

通過數學讓生活更美好!與Polly Ethylene教授一起分享。

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B2 中高級 中文 無限 術語 序列 條款 函數 面積

【用英文學數學】無窮級數 (Infinite Series)

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