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38

I have no evidence to back this up, nor do I know how I could obtain such a thing. But I strongly believe this is a vocabulary issue, and I would like to see the term "series" phased out of usage in this context. To many minds, "sequence" and "series" convey the same thing: a list of items. In modern parlance, we speak of a television series (a sequence of ...


22

I second Andrew Sanfratello's answer—there's just no getting around the fact that technical terms have technical meanings which differ from their everyday meanings. And when I teach series, I start each class by writing "SERIES MEANS SUM" at the top of the board every day. (At one previous institution, I often taught in a room with several large ...


20

In my experience, one of the problems with series is that usually you have two sequences if you investigate the series $\sum(a_n)$: the sequence $(a_n)$, and the sequence of partial sums $S_n=a_1+\ldots + a_n$. I noticed that trying to stress this distinction helps a lot. To the intuition, I like R. Péter: Playing with Infinity, the chocolate bar example on ...


15

Mathematics is littered with terms that are commonly used in the real world that can mean very different things. Sequence and series are just two. Group, function, kernel, field, and ring are just some examples of terms that have very exact meanings in various mathematical areas, but are used colloquially in English very differently. If you can emphasize ...


15

I tutored a student who came with a kind of problem I had never seen before and found quite refreshing. It was something like: A child is being pushed on a swing by their father, reaching a maximum height of 4 feet. The father stops pushing, and the maximum height of the swing decreases by 15% on each successive swing. I don't remember the question ...


13

(I have not done this exact presentation, so I cannot vouch for its efficacy. But I have used the main idea before, and it seems to help some students, and is at least a bit of fun. Also, this is meant to address the intuition aspect of the question, not the motivation.) Demonstrate by walking! Convergence means there's a spot that will be approached to any ...


12

It seems to me that the reason for learning series in calculus is to analyze functions through power series representations. There are other reasons series are important in mathematics, but in the standard calculus course, it seems to me that series are introduced for the sake of explaining power series. The key is then, Can one motivate power series ...


12

The Fourier coefficients contain useful information that is not at all apparent just from the shape of the periodic function itself. For example, when Fourier series are used for musical vibrations we may remove the terms in the Fourier series with coefficients having magnitude below a certain cutoff (kind of like taking out fundamental tones in a musical ...


10

Historically it seems that "sequence" is the interloper. "Series" was used to denote both concepts going as far back as Wallis, usually with the qualifier "infinite"; sometimes "progression" is used to denote what we call a sequence. Our current use of "sequence" in the theory of series did not become common until later, perhaps around 1900, as did the ...


10

I suggest that this is more than a problem in wording. In mathematics, many objects are introduced as a process (e.g. functions as "making a y out of a given x", sets as "taking things together") but later need to be used as an object (e.g. derivation of functions, sets of sets where the "inner" set must be perceived as an object (topology, measures)). This ...


9

Here's one you've probably seen. But I'll post it since I like it a lot. $\sum_{k=0}^{\infty}(1/\phi^2)^k=\phi$ where $\phi$ is the golden mean $1+(\sqrt{5}-1)/2$, approximately 1.618


9

The classic examples of the first $n$ natural numbers, first $n$ even numbers, and first $n$ odd numbers are all nice introductions. Here are some pictures I created to help explain them using blocks: Natural numbers: Even numbers: Odd numbers: Similar to the binary example given earlier, here is one using ternary (base 3): $$0.020202\ldots = \sum_{n = ...


9

As an engineering student, I dealt with binary all the time. A good calculator would convert from decimal to binary, hex, octal. When converting 1/3 to binary, I noted that it's .01010101... Which if you are used to reading binary past the decimal point is simply 1/4+1/16+1/64.... This seemed interesting to me.


9

Unfortunately, my example is not a full answer to your question, but I think, it helps students to be at least beware of what can happen. You can explain series as a inifite summation of areas (at least as long as everything is non-negative). And convergence of a series means: Is there a big rectangle such that all the given areas fit in? Intuitively, most ...


8

Another point that oughtn't be neglected was that, historically, numerical computation/approximation used Taylor-Maclaurin expansions (not to mention Newton-Raphson when convenient) to approximate root-taking. Newton was apparently very happy with his discovery of the binomial expansion for general exponents, although surely not only for numerical purposes. ...


8

It is unfortunate the term convergent has radically different meaning as it applies to sequences and series. A sequence $a_n \rightarrow L$ as $n \rightarrow \infty$ essentially means we can capture the tail of $a_n$ within as small an epsilon band as we desire. Whereas the series formed from $a_n$ converges iff the sequence of partial sums converges. Of ...


8

Here are a few more examples: the amount on your savings account ; the amount of money in your piggy bank if you deposit the same amount each week (a bank account with regular deposits leads you to arithmetico-geometric sequences) ; the size of a population in exponential growth, e.g. bacteria in a Petri dish (or in your leftovers if you find Petri dishes ...


8

I like to explain why arithmetic and geometric progressions are so ubiquitous. Using the examples other people have given. Geometric progressions happen whenever each agent of a system acts independently. For example population growth each couple do not decide to have another kid based on current population. So population growth each year is geometric. Each ...


7

There are tons of reasons: Signal theory: The key phrase here is bandwidth. If you want to transmit a signal over an analog channel you could modulate it onto a carrier frequency. To ensure that you do not have inference with other signals transmitted simultaneously, you should ensure that the signal you want to transmit does not have too large frequency ...


6

I like $41+\sum_{k=0}^{0}2k$, $41+\sum_{k=0}^{1}2k$, $41+\sum_{k=0}^{2}2k$, … through $41+\sum_{k=0}^{40}2k$ Each sums to a prime except for the 41st sum which is $41^2$ Attached is a pic of this sequence of primes:


6

Decimal versions of fractions give very simple examples. For example 1/3 = .333333333….. And what is that decimal but $\sum_{k=1}^{\infty}3/(10^k)$ Or 1/11 is .09090909... Which is $\sum_{k=1}^{\infty}9/(100^k)$


6

You cited the square wave, saying that it can be written as a piecewise function. True, but when given a box of resistors, capacitors, and inductors, you can't build a piecewise filter. This series is taught in engineering school and is one of the most beautiful examples of a Fourier series being used in real life. A filter creating a square wave that can ...


6

I agree that the form $\frac{r^n-1}{r-1}$ is in many respects more natural: certainly it is in cases where $r>1$, but even when $r$ is left unspecified, I think this form arises most naturally from many "customary" ways of deriving the formula. So why is it usually taught the other way around? I suspect the answer is precisely that, in most secondary ...


6

This is a nice demonstration that $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\cdots = 2 \;.$$                     (Image from Wikipedia article on Geometric progression.)


6

I believe option #2 (teach both and methods for choosing) is best. I'll try to illustrate with examples. $\displaystyle{\sum_{n=1}^\infty \frac{n}{n^2+1}}$ The intuition (which most students see quickly) is that this series is "pretty much" the same as $\sum \frac{1}{n}$ because the terms being added approach $\frac{1}{n}$ "in the long run" (as $n\to\infty$...


6

You should do option (2). As you already mentioned, there are series that cannot be done through limit comparison and can only be done with direct comparison. Why would you purposely leave those out? (OK, I can think of a few reasons, but they all lead me down the rabbit hole that is "What's the point of Calculus II?") Anyway, from a mathematical point of ...


5

Sequence is just a function of the type $f:\mathbb{N} \to \mathbb{R}$. It is common to list the elements of this sequence as $$(a_1,a_2,a_3,\ldots,a_n)\,.$$ One example is the sequence of all even numbers: $(0,2,4,6,8,10,\ldots)$. However some sequences may be defined in a different form when there is no easy formula for expressing the terms, like the ...


5

As a financial practitioner where such calculations are commonplace, I would encourage you to teach both and motivate them. In finance an annuity is a stream of regularly spaced payments. An "ordinary" annuity, e.g. a mortgage, has the first payment incurred at the beginning of the first period but paid at the end of the first period, so it includes ...


5

Another big advantage of direct comparison (if you have the time) is that it can be used to establish error estimates. For instance, if I want to estimate $$ \sum_1^\infty \frac{n+1}{n^4+n+7} $$ to within $\frac{1}{100}$. Then I can make the estimates $$ \frac{n+1}{n^4+n+7} \leq \frac{2n}{n^4} = \frac{2}{n^3} $$ Now, by drawing a picture we can see ...


4

Good answers!! What I had in mind was this: Fractals, as the Koch snowflake: https://en.wikipedia.org/wiki/Koch_snowflake The area is a convergent geometric series, while the circumference is infinite, represented by a divergent geometric series. The geometric series arises in such problems because of the self-similarity of the figure.


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