## 字幕列表 影片播放

• 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&quot;一&quot;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&#39;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&#39;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。

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&quot;極限&quot;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&quot;條款&quot;。

• 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&quot;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&quot;序列&quot;。

• 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&quot;a&quot;，用一個小標&quot;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，&quot;a&quot;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&#39;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，&quot;系列&quot;只是一個&quot;序列&quot;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&quot;求和符號&quot;。

• 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&quot;summation&quot;是大寫的希臘字母&quot;sigma&quot;。

• 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&quot;無限&quot;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&#39;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&#39;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&quot;收斂&quot;為一。

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

換句話說，這個系列是&quot;收斂&quot;。

• 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&#39;。

• 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&quot;分歧&quot;。

• 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。