シャドーイング練習: Convergence and Divergence: The Return of Sequences and Series - 動画で英語スピーキングを学ぶ

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Professor Dave here, let’s converge a little.

Earlier is this course, we introduced the concept of a sequence as a particular list of numbers, and we looked at some basic types, like arithmetic sequences, geometric sequences, and even some more interesting ones like the Fibonacci sequence.

We saw how these can be expressed in a few different ways, as beyond simply listing the terms, they can be abbreviated using a particular expression or formula.

We also expanded on this to look at series and summation notation.

If you’ve never seen these things before, go back and check out this tutorial now, because we are going to be building on that knowledge over the next few tutorials, since we now understand calculus.

First, we want to be able to assess whether a sequence is convergent or divergent, so let’s find out what that means.

Take the sequence A sub N equals N. That means that the first term is one, then two, and so forth through the integers.

We want to find out what this sequence does in the limit of N approaching infinity.

For this one it’s easy to see that the sequence approaches infinity as N approaches infinity, so we would call this sequence divergent, because the limit does not exist.

Then let’s look at something like A sub N equals N over N plus one.

This will give us one half, two thirds, three fourths, and so forth.

We can pretty quickly see that this sequence will approach one but never quite get there, so the limit of this sequence as N approaches infinity, is one.

Because the limit exists, as a finite number, we would call this sequence convergent.

There are some sequences for which it’s a little trickier to determine the limit, and with these it may be useful to apply some of the limit laws we already know, as they could help us manipulate things into a form where the limit is easier to assess.

Sometimes we can use special techniques, like using L’Hospital’s rule on the related function, or something called the squeeze theorem.

This one says that if there are three sequences, A, B, and C, and B is always between A and C, then if A and C both have a limit of L, then B must also have a limit of L, because it is being squeezed in between these ones.

Some sequences will be divergent without going to infinity, like negative one to the N power.

This just alternates between positive and negative one forever, so there is no limit.

So once again, we look at what a sequence does as N approaches infinity.

If the sequence asymptotically approaches some finite number, it is convergent.

If it does not do that, it is divergent.

Now that we understand convergence and divergence for sequences, let’s apply this understanding to series.

First we have to recall what a series is.

When we try to add up all the terms in an infinite sequence, we get an infinite series.

This can be represented by listing out all the terms, but it can also be abbreviated using the summation notation we already learned, with the upper case sigma.

We can talk about convergent and divergent series just like we did for sequences.

Starting again with a simple example, simply N, we can easily see that if we were to add up all the positive integers, which themselves go to infinity, certainly their sum will also be infinite.

This can be demonstrated by forming a new sequence, where the first term is the first term of the series, and then the sum of the first two terms of the series, and then the first three terms, and four terms, and so forth.

The limit of this sequence is the sum of the original series, and the sequence will go to infinity, so the original series is clearly divergent.

Can any infinite series be convergent?

That is, can an infinite number of terms add up to give a finite number?

Well when we looked at improper integrals, we saw exactly this sort of thing happening, with a limit of integration going to infinity but still yielding a finite value for the area under the curve, so we shouldn’t be too surprised that convergent infinite series exist.

Take for example one over two to the N. Listing the first few terms we get one half, one fourth, one eighth, and so on, approaching zero.

Let’s use the same technique as before, creating a new sequence and listing the first term of the series, then the sum of the first two, which is three fourths, then the sum of the first three, which is seven eighths, and if we continue on in this manner, we see that this sequence approaches one.

Therefore the sum of this series is equal to one, and we can say that it is convergent.

Different types of series will have different requirements for convergence.

Take this generalized geometric series.

If we recall the definition of a geometric sequence, it is one in which you multiply each term by some constant to get the next one, so starting with A, we get AR, then ARR, or AR squared, then AR cubed, and so forth.

So a geometric series is just the sum of these terms, or the sum of A times R to the (N minus one).

We will find that if the absolute value of R is greater than one, a geometric series will be divergent, because these terms will get bigger and bigger, and therefore go to infinity.

If equal to one, we are just adding up infinitely many A terms, which will also be divergent.

But going back to the sequence, if the absolute value of R is less than one, as it is raised to bigger and bigger exponents, each term will get smaller, and the limit of the sequence will be zero, so the series, or the sum of the terms, will be convergent, and the sum will be equal to A over the quantity (one minus R).

In fact, an important theorem states that in order for a series to be convergent, the sequence that generates the terms in the series must have a limit of zero.

This makes sense, because if the limit were any other number, or if the sequence itself was divergent, there would be no way for the terms in the series to add up to a finite value.

And even if the limit of the sequence is zero, it is still not a guarantee that the related series will converge.

So sometimes, we can look at a series and state that it is divergent simply by showing that the sequence it is derived from does not converge to zero.

Take something like N squared over the quantity (five N squared plus four) from one to infinity.

To assess the series, let’s just look at the sequence of numbers we get from this expression, without considering their sum.

What is the limit of this sequence as N approaches infinity?

Well we need some algebraic manipulation first.

Let’s divide both top and bottom by N squared so that we aren’t plugging in so many infinities.

On top we get one, and on the bottom we get five plus four over N squared.

Now we plug in infinity and this term goes to zero, leaving us with one fifth.

The sequence does not converge to zero, so the sum of this series that includes all the terms from that sequence must be divergent.

Now that we understand the concepts of convergence and divergence as they apply to both sequences and series, we are ready to move forward and learn more about series.

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このレッスンの語彙とスピーキングのポイント

この動画には、シャドーイング用の文が55文、単語が1296語あります。音声の長さは9:07です。話す速さは1分あたり約142語で安定しており、シャドーイングしやすいペースです。英語の頻出3,000語に含まれる単語は79%だけなので、語彙は難しめです。

この動画の重要語彙

動画に出てくる覚えておきたい単語15語を、発音と意味つきで紹介します。

単語	発音	意味
sequence 名詞	/ˈsiː.kwəns/	順序
infinity 名詞	/ɪnˈfɪnɪti/	無限
infinite 形容詞	/ˈɪnfɪnɪt/	途方もない, 巨大な
finite 形容詞	/ˈfaɪ.naɪt/	限られた, 有限な
technique 名詞	/tɛkˈniːk/	技術
recall 動詞	/ɹɪˈkɔl/	呼び返す
eighth 形容詞	/eɪtθ/	八番目
plug 名詞	/plʌɡ/	プラグ
quantity 名詞	/ˈkwɑn(t)ɪti/	量, 数量
minus	/ˈmaɪ.nəs/	引く
squeeze 動詞	/skwiːz/	押す, 締め付ける
tutorial 名詞	/ˌtjuːˈtɔːɹɪəl/	学習用, チュートリアル
theorem 名詞	/ˈθiərəm/	定理
formula 名詞	/ˈfɔː.mjʊ.lə/	公式, 定式
guarantee 名詞	/ˌɡæɹ.ənˈtiː/	保証

動画に出てくる句動詞

単語	意味
find out 動詞	見出す, 調べる
check out 動詞	試験する, 検査する
go back 動詞	戻る, 帰る

この動画の文法

話し手がよく使っている文型を、動画の実際の表現とともに紹介します。

文型	動画での表現
受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる	can be expressed · being squeezed · can be represented
関係詞節 who / which + 節 — 人や物について情報を加える	two, which is · terms, which will

注意したい発音

話し手はaren't, shouldn't, you'veなど、短縮形や弱形を3回使っています。聞こえたとおりの短い形で発音しましょう。

「th」の音: eighth /eɪtθ/, theorem /ˈθiərəm/, arithmetic /æɹɪθˈmɛtɪk/
「sh」と「zh」の音: integration /ˌɪn.tɪˈɡɹeɪ.ʃən/, manipulation /məˌnɪp.juˈleɪ.ʃən/, notation /noʊˈteɪʃən/, summation /sʌˈmeɪʃən/
長い単語（アクセントの位置に注意）: infinity /ɪnˈfɪnɪti/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, integration /ˌɪn.tɪˈɡɹeɪ.ʃən/, manipulation /məˌnɪp.juˈleɪ.ʃən/, manipulate /məˈnɪp.jʊ.leɪt/

この動画での練習方法

まず声を出さずに動画を最後まで聞き、知らない単語をメモします。
通常の速度で一文ずつシャドーイングし、話し手のリズムに合うまで繰り返します。
自分の声を録音して元の音声と比べます。sequence, infinity, infiniteなどの単語に特に注意しましょう。

シャドーイングとは？英語上達に効果的な理由

シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。

シャドーイングのやり方: ステップ別の完全ガイドを読む →

シャドーイング練習: Convergence and Divergence: The Return of Sequences and Series - 動画で英語スピーキングを学ぶ

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