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

或

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!