Pratica di Shadowing: Convergence and Divergence: The Return of Sequences and Series - Impara a parlare inglese con i video
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Professor Dave here, let’s converge a little.
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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.
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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.
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We also expanded on this to look at series and summation notation.
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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.
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First, we want to be able to assess whether a sequence is convergent or divergent, so let’s find out what that means.
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Take the sequence A sub N equals N. That means that the first term is one, then two, and so forth through the integers.
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We want to find out what this sequence does in the limit of N approaching infinity.
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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.
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Then let’s look at something like A sub N equals N over N plus one.
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This will give us one half, two thirds, three fourths, and so forth.
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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.
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Because the limit exists, as a finite number, we would call this sequence convergent.
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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.
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Sometimes we can use special techniques, like using L’Hospital’s rule on the related function, or something called the squeeze theorem.
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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.
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Some sequences will be divergent without going to infinity, like negative one to the N power.
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This just alternates between positive and negative one forever, so there is no limit.
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So once again, we look at what a sequence does as N approaches infinity.
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If the sequence asymptotically approaches some finite number, it is convergent.
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If it does not do that, it is divergent.
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Now that we understand convergence and divergence for sequences, let’s apply this understanding to series.
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First we have to recall what a series is.
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When we try to add up all the terms in an infinite sequence, we get an infinite series.
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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.
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We can talk about convergent and divergent series just like we did for sequences.
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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.
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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.
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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.
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Can any infinite series be convergent?
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That is, can an infinite number of terms add up to give a finite number?
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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.
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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.
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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.
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Therefore the sum of this series is equal to one, and we can say that it is convergent.
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Different types of series will have different requirements for convergence.
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Take this generalized geometric series.
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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.
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So a geometric series is just the sum of these terms, or the sum of A times R to the (N minus one).
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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.
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If equal to one, we are just adding up infinitely many A terms, which will also be divergent.
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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).
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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.
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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.
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And even if the limit of the sequence is zero, it is still not a guarantee that the related series will converge.
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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.
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Take something like N squared over the quantity (five N squared plus four) from one to infinity.
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To assess the series, let’s just look at the sequence of numbers we get from this expression, without considering their sum.
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What is the limit of this sequence as N approaches infinity?
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Well we need some algebraic manipulation first.
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Let’s divide both top and bottom by N squared so that we aren’t plugging in so many infinities.
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On top we get one, and on the bottom we get five plus four over N squared.
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Now we plug in infinity and this term goes to zero, leaving us with one fifth.
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The sequence does not converge to zero, so the sum of this series that includes all the terms from that sequence must be divergent.
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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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Vocabolario e note di pronuncia per questa lezione
Questo video contiene 55 frasi e 1296 parole da ripetere con lo shadowing. Il parlato dura 9:07. Chi parla mantiene un ritmo costante di circa 142 parole al minuto, comodo per lo shadowing. Solo il 79% delle parole rientra nelle 3.000 più comuni dell’inglese, quindi il lessico è impegnativo.
Vocaboli chiave di questo video
15 parole del video che vale la pena imparare, con pronuncia e significato:
| Parola | Pronuncia | Significato |
|---|---|---|
| sequence sostantivo | /ˈsiː.kwəns/ | sequenza |
| infinite aggettivo | /ˈɪnfɪnɪt/ | infinito |
| finite aggettivo | /ˈfaɪ.naɪt/ | finito |
| geometric aggettivo | /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/ | geometrico |
| assess verbo | /əˈsɛs/ | valutare, stimare |
| technique sostantivo | /tɛkˈniːk/ | tecnica |
| recall verbo | /ɹɪˈkɔl/ | ritirare, revocare |
| eighth aggettivo | /eɪtθ/ | ottavo, ottava |
| plug sostantivo | /plʌɡ/ | spina |
| quantity sostantivo | /ˈkwɑn(t)ɪti/ | grandezza, quantità |
| minus | /ˈmaɪ.nəs/ | meno |
| squeeze verbo | /skwiːz/ | spremere, stringere |
| tutorial sostantivo | /ˌtjuːˈtɔːɹɪəl/ | esercitazione |
| theorem sostantivo | /ˈθiərəm/ | teorema |
| formula sostantivo | /ˈfɔː.mjʊ.lə/ | formula |
I phrasal verb che sentirai
| Parola | Significato |
|---|---|
| add up verbo | sommarsi |
| find out verbo | scoprire |
| check out verbo | investigare |
| go back verbo | tornare |
La grammatica di questo video
Le strutture che chi parla usa di più, con le parole esatte del video:
| Struttura | Nel video |
|---|---|
| Forma passiva be + participio passato — conta ciò che accade, non chi lo fa | can be expressed · being squeezed · can be represented |
| Frasi relative who / which + frase — un’informazione in più su una persona o una cosa | two, which is · terms, which will |
Pronuncia a cui fare attenzione
Chi parla usa 3 contrazioni e forme ridotte, come aren't, shouldn't, you've. Pronunciale nella forma breve, così come le senti.
- I suoni “th”: eighth /eɪtθ/, theorem /ˈθiərəm/, arithmetic /æɹɪθˈmɛtɪk/
- I suoni “sh” e “zh”: integration /ˌɪn.tɪˈɡɹeɪ.ʃən/, manipulation /məˌnɪp.juˈleɪ.ʃən/, notation /noʊˈteɪʃən/, summation /sʌˈmeɪʃən/
- Parole lunghe — attenzione all’accento: infinity /ɪnˈfɪnɪti/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, integration /ˌɪn.tɪˈɡɹeɪ.ʃən/, manipulation /məˌnɪp.juˈleɪ.ʃən/, manipulate /məˈnɪp.jʊ.leɪt/
Come esercitarsi con questo video
- Ascolta tutto il video una volta senza parlare e annota le parole che non conosci.
- Fai shadowing frase per frase a velocità normale, ripetendo ognuna finché il tuo ritmo coincide con quello di chi parla.
- Registrati e confronta con l’originale, facendo attenzione a parole come sequence, infinite, finite.
Cos'è la tecnica dello Shadowing?
Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.