Name: 【用英文學數學】無窮級數 (Infinite Series)
Uploaded: 2013-01-18T12:00:20.000Z
Duration: 14 min 57 s

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。

an infinite number of them will always be infinity.

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.

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。

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。

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。

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

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？

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.

Then you walk half of the remaining distance

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!

讓我們以分數的十分之九為例。

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

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。

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。

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"。

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"

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?

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。

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。

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

我們稱之為"a"，用一個小標"n"編號。

which tells us which term in the sequence it is

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

所以，比如第二十項，"a"次二十項。

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。

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"。

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

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.

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。

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【用英文學數學】無窮級數 (Infinite Series)

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