4.5 Sequences and Series: Mathematics Communication
So far, you have focused on written mathematical language. In this section, you will practise understanding and taking part in spoken mathematical discussions. You will read and analyse two different dialogues: one between two students, and one between a student and a lecturer.
The goal of this section is not just to understand these dialogues, but to help you feel confident in having similar discussions yourself. You will learn how to have a discussion about mathematics, ask and answer questions, and explain your ideas clearly to someone else. This will help you use English for practical conversations about mathematics, both in class and out of class. To do this, you will need to:
-
notice useful phrases and sentence structures in the dialogues,
-
memorise them,
-
practise saying them,
-
and finally, use them in real conversations.
Each line in the dialogues is numbered to help you talk about specific parts easily. For example, if you have a question, you can say: What does the second word from the end of line 8 mean?
By working with these dialogues, you will improve your ability to discuss mathematics in English, ask and answer questions, and communicate your ideas clearly with others.
Dialogue 1: Two students
An example dialogue between two university students, Elise and Sam, discussing sequences and series. This dialogue occurs in a study space with the two students sitting close to each other.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Elise: | Hi Sam, I’ve been working on the sequences and series homework, but I’m struggling with some terminology. Can you please help me? |
| (2) | Sam: | Sure, Elise! What part are you finding tricky, exactly? |
| (3) | Elise: | Well, I’m trying to figure out if this sequence converges or diverges, but I’m not entirely sure what these terms mean. I get that they’re opposite of each other, that’s kind of obvious from the language… |
| (4) | Sam: | The main idea is that a sequence converges if it approaches a specific value, which we call the limit, as n approaches infinity. Think of the limit as a target number. If the terms actually settle close to that number, then that’s the limit and the sequence converges. On the other hand, if the terms don’t approach any particular value, we say the sequence diverges. |
| (5) | Elise: | I get it now… So, can you give me an example of a sequence that converges and a sequence that diverges? |
| (6) | Sam: | Alright, let’s think. Well, the sequence one over n converges to zero since, as n approaches infinity, one over n gets closer and closer to zero. But the sequence n squared diverges, since as n approaches infinity, n squared grows without bound. |
| (7) | Elise: | Yeah, that makes sense. Finally! Now, please remind me again, what’s the difference between a sequence and a series? I keep getting them mixed up. |
| (8) | Sam: | That’s a pretty common confusion. A sequence is just a list of terms. A series, however, is the sum of the terms of a sequence. So, if you add up all the terms, you get a series. |
| (9) | Elise: | Got it. So our homework problem is a series. We need to find the sum from n equals one to infinity, of one over, two to the power n. |
| (10) | Sam: | That’s right. This is actually an infinite geometric series. And we have a formula for this. Do you remember it? |
| (11) | Elise: | Let me check my notes… OK so the sum is a over, one minus r, where a is the first term and r is the common ratio, provided that the absolute value of r is less than one. |
| (12) | Sam: | Great! So, what is the first term and what is the common ratio in our example? |
| (13) | Elise: | The first term is one half, and the common ratio is also one half. |
| (14) | Sam: | Excellent. And since the absolute value of one half is less than one, the series converges, which means we can apply the formula. |
| (15) | Elise: | So, the sum would be one half, divided by, one minus one half. Which is just one. |
| (16) | Sam: | You’ve got it. Series can be a tricky topic, but once you understand the terminology and the formulas, they become much easier. |
| (17) | Elise: | Thanks so much, Sam. Everything is so much clearer now! |
| (18) | Sam: | No worries, Elise. Anytime! Have a great day! |
| (19) | Elise: | And you. Bye for now. |
Activity 1: Global understanding
From the options below, select the summary that you think most accurately describes the broad structure of lines (1) to (8) of the conversation. Next, select the ‘Answer’ button below the options.
Option 1: Elise first asks Sam to clarify the difference between two terms, which Sam does by supplying definitions and then examples of each. Elise then asks for clarification for two further terms, which Sam explains using brief definitions.
Option 2: Elise first asks Sam to clarify the difference between two terms, which Sam does by supplying two examples of each and a detailed counter-example. Elise then asks for clarification for two further terms. Sam explains these using definitions, that he admits are highly complex.
Option 3: Elise first asks Sam to clarify the difference between two terms, which Sam does by supplying two examples of each and a single counter-example. Elise then asks for clarification for three further terms. Sam explains these by providing one example and applying it to each of the three terms.
Option 4: Elise first asks Sam to clarify the difference between two terms, which Sam does by supplying two brief definitions of each and an example of each. Elise then immediately asks Sam for help with her homework, which Sam does by referring to the first pair of terms.
Answer
Option 1 is the correct answer. The first pair of terms for which Elise asks for clarification are converge and diverge in (3). Sam provides definitions in (4), and examples in (6). The second pair of terms for which Elise asks for clarification are sequence and series in (7). Sam responds with two short definitions in (8). The conversation then moves to the homework.
Option 2 is incorrect because Sam offers no counter-examples. Neither does Sam state that the definitions for the second pair of terms are complex.
Option 3 is incorrect because Sam offers no counter-example. Moreover, Emma’s second request for help is about a second pair of terms, not a set of three terms.
Option 4 is incorrect because Elise’s request for clarification regarding the homework a) follows a question about the second pair of terms and b) falls outside lines (1) to (8) of the dialogue.
Activity 2: Identifying useful language
Question 1: The conversation you have listened to is a fairly informal dialogue between two students. This activity focuses on some of the informal language that is used. Identify useful phrases in the dialogue using the following prompts:
- Between (1) and (5), where does Elise say something that means, in this context, something like the form of the terminology itself makes it evident?
- Where does Elise say a single word sentence that means something like I understand this at last and I was getting frustrated about not having understood it?
- Identify three other places where Elise says something that means I understand now but perhaps without the element of frustration in 2.
Next, select the ‘Answer’ button below.
Answer
- In (3), that’s kind of obvious from the language
- Finally! (7)
- I get it now (5).Yeah, that makes sense (7). Got it (9).
Question 2: When studying mathematics, it is very common and entirely normal to encounter confusion. This exercise makes two points about the language of confusion before exploring some other phrases about this common experience. Firstly, in (8) Sam says That’s a pretty common confusion. There are two points to make here, first about structure and secondly about usage. As a first point, note that pretty here means quite. We might say That’s a pretty challenging problem which means the same as That’s quite a challenging problem (note that the article a is in a different place relative to quite and pretty). As a second point, admitting confusion and asking for help is an important and common part of studying mathematics. Notice how Elise openly states that she is confused in order to get the help she needs. Identify five phrases in the dialogue which are about confusion. Next, select the ‘Answer’ button below.
Answer
- Struggling with some terminology (1)
- finding X tricky (2)
- trying to figure X out (3)
- I’m not entirely sure what these terms mean (3)
- keep getting X mixed up (7).
As a follow-up activity, you might wish to reflect on your own journey through mathematics and on what you have found confusion (you could of course compare this with anything that you have found obvious, easy, straightforward or clear). Ask yourself how you have handled the confusions that you encountered? What ways have you found to manage and to move through these confusions? What alternative ways of working with mathematical confusion have you not yet tried? How familiar are you with the language of confusion and with talking about confusion with fellow students and with lecturers.
Language Focus
In (3), Elise says I’m trying to figure out if this sequence converges or diverges. The string figure out is a phrasal verb. Phrasal verbs often consist of a verb (here figure) and a particle (here out). There are several phrasal verbs which use out which are common in mathematics. Match the examples below with their meaning. Next, select the ‘Answer’ button below.
| Phrasal verb example | Meaning | |
|---|---|---|
| Please make sure you set out the problem very clearly in this exam. | To solve or answer something [1]. This phrasal verb is quite neutral in style. | |
| We will need a decent Python code to work out the answer to this question. | To format a document in terms of its overall appearance. | |
| A dissertation should be laid out very clearly with a contents page, list of figures and standardised headings. | To present something well so it can be easily read. | |
| I need to some help to figure out the answer to this seemingly impossible problem. | This phrasal verb forms part of an idiom the whole of which means to unexpectedly produce something. | |
| I thought about the problem for ages and got nowhere. But then. somehow, I just pulled the answer out of thin air. | To solve or answer something [2]. this phrasal verb is informal and may imply that you have been puzzling about what you want to solve. |
Answer
| Phrasal verb example | Meaning |
|---|---|
| Please make sure you set out the problem very clearly in this exam. | To present something well so it can be easily read. |
| We will need a decent Python code to work out the answer to this question. | To solve or answer something [1]. This phrasal verb is quite neutral in style. |
| A dissertation should be laid out very clearly with a contents page, list of figures and standardised headings. | To format a document in terms of its overall appearance. |
| I need to some help to figure out the answer to this seemingly impossible problem. | To solve or answer something [2]. this phrasal verb is informal and may imply that you have been puzzling about what you want to solve. |
| I thought about the problem for ages and got nowhere. But then. somehow, I just pulled the answer out of thin air. | This phrasal verb forms part of an idiom the whole of which means to unexpectedly produce something. |
There are two further things to note. Firstly, that the final example pull the problem out of thin air is closer to an idiom because it is the whole phrase (including the string of thin air) that should be learned and used together, not just pull out by itself. Note secondly that there are two paraphrases of the form to solve or answer something. One is more formal and one is more informal.
Note that the first four phrasal verbs can have several different structures as in the table below. In the first column, the particle is adjacent to the verb, with the object noun phrase coming after the particle. In the second column, the object noun phrase is between the verb and the particle. This is often called the separable form of the phrasal verb. In the third column, where the object is a pronoun, only the separable form can be used, as in the second column.
| Verb + particle + object noun phrase | Verb + object noun phrase + particle (separable form) | Verb + pronoun + particle |
|---|---|---|
| Set out the problem | Set the problem out | Set it out (not |
| Work out the answer | Work the answer out | Work it out (not |
| Lay out my dissertation | Lay my dissertation out | Lay it out (not |
| Figure out the answer | Figure the answer out | Figure it out (not |
Phrasal verbs are a complex part of English grammar and not all phrasal verbs are separable. However, the structures shown above are correct for the verbs listed. We encourage you to look around for other phrasal verbs and to notice what they mean, how they are used and whether they are separable or not.
Activity 3: Your turn!
Goal
Practise having a conversation about resolving a set of confusions about series and sequences in English. (We might even say a series of confusions about series – which is a gentle mathematics joke!) Focus on communication, not necessarily on solving new math problems. Working together, create a set of confusions that you have or have had and use these to construct the dialogue.
Set-up
In pairs, take the following roles:
Speaker A is the student who has a set of confusions about series or sequences. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don’t.
Speaker B is the student that has the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles – one student takes Speaker A and the other student takes Speaker B.
Step 2: Decide on a set of confusions that you have or have had about sequences or series. These might be about terminology, concepts, methods, definitions, examples, application or something else.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any useful phrases.
Step 4: Run the dialogue. Speaker A should start, either with a variation on Elise’s opening above, or with something else, for example, Hi [Speaker B]. I’m not sure how to approach [Problem P]. Do you have a moment to help?
Step 5: Switch roles and practise again to experience both asking and answering.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced, how confident you felt, and if there are alternative ways of going through the dialogue.
Dialogue 2: A student and a lecturer
An example dialogue between a university student, Jade, and her lecturer, Professor Brown, discussing Taylor series. The dialogue takes place in an office, during the lecturer’s office hours in a formal but welcoming setting. Jade enters the office and sits next to Professor Brown.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Jade: | Er, hello, Professor Brown. Are you available now at all? I am struggling with Taylor series. I’ve reviewed the notes several times, but I still find the concept quite confusing. |
| (2) | Professor Brown: | Of course, Jade, come in. What specifically are you finding difficult? |
| (3) | Jade: | Well, I understand that a Taylor series is a way to represent a function as an infinite sum, but I don’t really understand why we would want to do that in the first place. |
| (4) | Professor Brown: | That’s a great question. Taylor series allow us to approximate complicated functions using polynomials, which are much easier to work with. Let’s start with the big picture: do you remember what the general form of a Taylor series is? |
| (5) | Jade: | I’ve actually learnt this! The Taylor series of a function f of x, expanded about a point a, is the sum from n equals zero to infinity, of the nth derivative of f evaluated at a, divided by n factorial, times x minus a, to the power n. |
| (6) | Professor Brown: | Perfect. Memorising the definition is an important first step, even when we don’t understand the whole concept. Let’s apply this by working through an example together. Can you tell me what the Taylor series for e to the power x would be, expanded about zero? |
| (7) | Jade: | Give me a moment; I know I know this! First, I need to find the derivatives of e to the power x. The first derivative is just e to the power x. Hence, the second derivative is also e to the power x, and actually, all the derivatives are e to the power x. |
| (8) | Professor Brown: | And what happens when we evaluate these derivatives at x equals zero? |
| (9) | Jade: | Let’s see… Since e to the power zero equals one, all the derivatives evaluated at zero equal one. |
| (10) | Professor Brown: | Exactly. So, what does the series become? |
| (11) | Jade: | The series would be the sum from n equals zero to infinity, of one divided by n factorial, times x to the power n. So that’s one, plus x, plus x squared over two, plus x cubed over six, and so on. |
| (12) | Professor Brown: | That’s right. Remember, we also need to consider for which values of x this series converges. |
| (13) | Jade: | I’m not sure how to determine that. |
| (14) | Professor Brown: | For that we can use the ratio test. For the exponential function, it turns out that this Taylor series converges for all real values of x. We therefore say that the radius of convergence is infinite. What do you think would happen if we only took the first few terms of the series instead of all infinitely many terms? |
| (15) | Jade: | Just a sec, give me a second to think if that’s OK. I suppose we would get an approximation of the function, rather than the exact value. |
| (16) | Professor Brown: | Exactly right. The more terms we include, the better our approximation becomes. The partial sum of the first n terms is called the nth degree Taylor polynomial. |
| (17) | Jade: | That makes so much sense. Thank you so much, Professor Brown. This has really helped me understand the concept. |
| (18) | Professor Brown: | You’re very welcome, Jade. Keep up the good work, and feel free to come back if you have any more questions. |
Activity 1: Global understanding
From the options below, select the summary that you think most accurately describes the overall structure of the dialogue. Next, select the ‘Answer’ button below the options.
Option 1: Jade has learnt a definition but she struggles to apply this to an example, and Professor Brown does this part for him. However, Jade is then able to generalise on her own from Professor Brown’s example to a second example.
Option 2: Jade has learnt a definition but it is not wholly correct. After modifying the definition, Jade is able to respond immediately to three different example problem that Professor Brown brings up.
Option 3: Jade has learnt a definition which she can then apply to an example with Professor Brown’s support. Finally, Professor Brown raises an additional complication which Jade works through again with Professor Brown’s support.
Option 4: Jade has learnt a definition and applies it to an example easily and fluently, without any support from Professor Brown. Jade then also raises a complication which he himself again addresses.
Answer
Option 3 is the correct answer. Jade gives the definition in (5). Between (6) and (11) Professor Brown supports Jade in applying that definition to one example. Finally, in (12), Professor Brown introduces a complication and Jade again works with Professor Brown to solve this.
Activity 2: Identifying useful language
Question 1: Jade is a typical student in the sense that there are some things she understands well and is confident about, while there are other things that she is less sure about. A good example of when Jade is sure about something is in (5): Jade has learnt a definition correctly. At other places, Jade is less sure of the next step. However, she handles this well in the dialogue by politely asking for some time to think about the next step. Identify three occasions where Jade says something that means something like I need a short amount of time to think this through. Next, select the ‘Answer’ button below.
Answer
Give me a moment (7); Let’s see (9); Just a sec [= second]; give me a second to think if that’s OK (15).
Question 2: Identify something Professor Brown says which means It’s a good idea to learn definitions even if we’re not wholly clear what every aspect of the definition might mean. Next, select the ‘Answer’ button below.
Answer
Memorising the definition is an important first step, even when we don’t understand the whole concept (6).
Question 3: In something Professor Brown says, locate a six-word phrase that means something similar to Let’s zoom out here first of all or Let’s think about things from a bird’s eye view or the macro-level to start off with. Next, select the ‘Answer’ button below.
Answer
Let’s start with the big picture (4). The collocation big picture is a common English phrase suggesting a top-level, overall perspective on a subject; one that is not necessarily fully detailed. It differs slightly from comprehensive, which implies both breadth and detail; for example, a comprehensive report on the first-year marks for the algebra exam suggests thoroughness across all aspects. Big picture can also function as a compound modifier before a noun, in which case it is hyphenated: Taking a big-picture approach to this problem. In this usage it is an adjective phrase, not a noun phrase. In mathematics, we often choose between two broad approaches. Sometimes we begin with the big picture where we consider a problem from a general, high-level perspective. On other occasions, we take a different approach: we begin with a single specific example and move progressively outwards, adding further examples until a general pattern or result emerges. This second approach is characteristic of inductive reasoning.
Activity 3: Your turn!
Goal
Practise having conversations about mathematics problems regarding sequences and series in English. Focus on communication, not necessarily on solving new math problems.
Set-up
In pairs, take the following roles:
Speaker A is the student with a question about the mathematics of sequences and series. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don’t.
Speaker B is the lecturer, professor or seminar leader with the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles – one student takes Speaker A and the other student takes Speaker B. Think about how this student-lecturer dialogue might be different to the student-student dialogue in the beginning of the section. The lecturer is highly likely to know the answers to the problems but might also ask helpful questions or provide partial examples in order to guide the student.
Step 2: Decide a problem from the topic of sequences and series to discuss.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any phrases that you feel may be useful.
Step 4: Run the dialogue. Speaker A should start, either with a variation on Jade’s opening utterance above, or with something else, for example, Excuse me Professor Brown, would you happen have some time to help me out with [Problem P]. It shouldn’t take too long.
Step 5: Run the dialogue a second time with the roles reversed.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced; how confident you felt; and if there are alternative ways of going through the dialogue.
In this section, you practised reading and performing dialogues (spoken communication) about mathematics. In the next section, you will focus on written mathematical communication.
