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Wed, 23 Sep 2020
The mystery of the malformed command-line flags
Today a user came to tell me that their command
failed, saying:
This is surprising. The command looks correct. The branch name is
required. The The
But it still didn't work:
I dug in to the script and discovered the problem, which was not actually a programming error. The crappy shell script was behaving correctly! I had written up release notes for the In an earlier draft of the release notes, when they were displayed as an HTML page, there would be bad line breaks:
or:
No problem, I can fix it! I just changed the pair of hyphens ( But then this hapless user clipboard-copied the option string out of
the release notes, including its U+2011 characters. The parser in the
script was (correctly) looking for U+002D characters, and didn't
recognize One lesson learned: people will copy-paste stuff out of documentation, and I should be prepared for that. There are several places to address this. I made the error message more transparent; formerly it would complain only about the first argument, which was confusing because it was the one argument that wasn't superfluous. Now it will say something like
which is more descriptive of what it actually doesn't like. I could change the nonbreaking hyphens in the release notes back to regular hyphens and just accept the bad line breaks. But I don't want to. Typography is important. One idea I'm toying with is to have the shell script silently replace all nonbreaking hyphens with regular ones before any further processing. It's a hack, but it seems like it might be a harmless one. So many weird things can go wrong. This computer stuff is really complicated. I don't know how anyone get anything done. [ Addendum: A reader suggests that I could have fixed the line breaks with CSS. But the release notes were being presented as a Slack “Post”, which is essentially a WYSIWYG editor for creating shared documents. It presents the document in a canned HTML style, and as far as I know there's no way to change the CSS it uses. Similarly, there's no way to insert raw HTML elements, so no way to change the style per-element. ] [Other articles in category /prog/bug] permanent link Sun, 13 Sep 2020The front page of NPR.org today has this headline: It contains this annoying phrase:
Someone has really committed to hedging. I would have said that the race would certainly help determine control of the Senate, or that it could determine control of the Senate. The statement as written makes an extremely weak claim. The article itself doesn't include this phrase. This is why reporters hate headline-writers. [Other articles in category /lang] permanent link Fri, 11 Sep 2020
Historical diffusion of words for “eggplant”
In reply to my recent article about the history of words for “eggplant”, a reader, Lydia, sent me this incredible map they had made that depicts the history and the diffusion of the terms: Lydia kindly gave me permission to share their map with you. You can see the early Dravidian term vaḻutanaṅṅa in India, and then the arrows show it travelling westward across Persia and, Arabia, from there to East Africa and Europe, and from there to the rest of the world, eventually making its way back to India as brinjal before setting out again on yet more voyages. Thank you very much, Lydia! And Happy Diada Nacional de Catalunya, everyone! [Other articles in category /lang/etym] permanent link
A maxim for conference speakers
The only thing worse than re-writing your talk the night before is writing your talk the night before. [Other articles in category /talk] permanent link Fri, 28 Aug 2020This morning Katara asked me why we call these vegetables “zucchini” and “eggplant” but the British call them “courgette” and “aubergine”. I have only partial answers, and the more I look, the more complicated they get. ZucchiniThe zucchini is a kind of squash, which means that in Europe it is a post-Columbian import from the Americas. “Squash” itself is from Narragansett, and is not related to the verb “to squash”. So I speculate that what happened here was:
The Big Dictionary has citations for “zucchini” only back to 1929, and “courgette” to 1931. What was this vegetable called before that? Why did the Americans start calling it “zucchini” instead of whatever they called it before, and why “zucchini” and not “courgette”? If it was brought in by Italian immigrants, one might expect to the word to have appeared earlier; the mass immigration of Italians into the U.S. was over by 1920. Following up on this thought, I found a mention of it in Cuniberti, J. Lovejoy., Herndon, J. B. (1918). Practical Italian recipes for American kitchens, p. 18: “Zucchini are a kind of small squash for sale in groceries and markets of the Italian neighborhoods of our large cities.” Note that Cuniberti explains what a zucchini is, rather than saying something like “the zucchini is sometimes known as a green summer squash” or whatever, which suggests that she thinks it will not already be familiar to the readers. It looks as though the story is: Colonial Europeans in North America stopped eating the zucchini at some point, and forgot about it, until it was re-introduced in the early 20th century by Italian immigrants. When did the French start calling it courgette? When did the Italians start calling it zucchini? Is the Italian term a calque of the French, or vice versa? Or neither? And since courge (and gourd) are evidently descended from Latin cucurbita, where did the Italians get zucca? So many mysteries. EggplantHere I was able to get better answers. Unlike squash, the eggplant is native to Eurasia and has been cultivated in western Asia for thousands of years. The puzzling name “eggplant” is because the fruit, in some varieties, is round, white, and egg-sized. The term “eggplant” was then adopted for other varieties of the same plant where the fruit is entirely un-egglike. “Eggplant” in English goes back only to 1767. What was it called before that? Here the OED was more help. It gives this quotation, from 1785:
I inferred that the preceding text described it under a better-known name, so, thanks to the Wonders of the Internet, I looked up the original source:
(Jean-Jacques Rosseau, Letters on the Elements of Botany, tr. Thos. Martyn 1785. Page 202. (Wikipedia)) The most common term I've found that was used before “egg-plant” itself is “mad apple”. The OED has cites from the late 1500s that also refer to it as a “rage apple”, which is a calque of French pomme de rage. I don't know how long it was called that in French. I also found “Malum Insanam” in the 1736 Lexicon technicum of John Harris, entry “Bacciferous Plants”. Melongena was used as a scientific genus name around 1700 and later adopted by Linnaeus in 1753. I can't find any sign that it was used in English colloquial, non-scientific writing. Its etymology is a whirlwind trip across the globe. Here's what the OED says about it:
Wowzers. Okay, now how do we get to “aubergine”? The list above includes Arabic bāḏinjān, and this, like many Arabic words was borrowed into Spanish, as berengena or alberingena. (The “al-” prefix is Arabic for “the” and is attached to many such borrowings, for example “alcohol” and “alcove”.) From alberingena it's a short step to French aubergine. The OED entry for aubergine doesn't mention this. It claims that aubergine is from “Spanish alberchigo, alverchiga, ‘an apricocke’”. I think it's clear that the OED blew it here, and I think this must be the first time I've ever been confident enough to say that. Even the OED itself supports me on this: the note at the entry for brinjal says: “cognate with the Spanish alberengena is the French aubergine”. Okay then. (Brinjal, of course, is a contraction of berengena, via Portuguese bringella.) Sanskrit vātiṅgaṇa is also the ultimate source of modern Hindi baingan, as in baingan bharta. (Wasn't there a classical Latin word for eggplant? If so, what was it? Didn't the Romans eat eggplant? How do you conquer the world without any eggplants?) [ Addendum: My search for antedatings of “zucchini” turned up some surprises. For example, I found what seemed to be many mentions in an 1896 history of Sicily. These turned out not to be about zucchini at all, but rather the computer's pathetic attempts at recognizing the word Σικελίαν. ] [ Addendum 20200831: Another surprise: Google Books and Hathi Trust report that “zucchini” appears in the 1905 Collier Modern Eclectic Dictionary of the English Langauge, but it's an incredible OCR failure for the word “acclamation”. ] [ Addendum 20200911: A reader, Lydia, sent me a beautiful map showing the evolution of the many words for ‘eggplant’. Check it out. ] [Other articles in category /lang/etym] permanent link Mon, 24 Aug 2020Ripta Pasay brought to my attention the English cookbook Liber Cure Cocorum, published sometime between 1420 and 1440. The recipes are conveyed as poems:
(Original plus translation by Cindy Renfrow) “Conyngus” is a rabbit; English has the cognate “coney”. If you have read my article on how to read Middle English you won't have much trouble with this. There are a few obsolete words: sere means “separately”; myed bread is bread crumbs, and amydone is starch. I translate it (very freely) as follows:
Thanks, Ripta! [Other articles in category /food] permanent link Fri, 21 Aug 2020
Mixed-radix fractions in Bengali
[ Previously, Base-4 fractions in Telugu. ] I was really not expecting to revisit this topic, but a couple of weeks ago, looking for something else, I happened upon the following curiously-named Unicode characters:
Oh boy, more base-four fractions! What on earth does “NUMERATOR ONE LESS THAN THE DENOMINATOR” mean and how is it used? An explanation appears in the Unicode proposal to add the related “ganda” sign:
(Anshuman Pandey, “Proposal to Encode the Ganda Currency Mark for Bengali in the BMP of the UCS”, 2007.) Pandey explains: prior to decimalization, the Bengali rupee (rupayā) was divided into sixteen ānā. Standard Bengali numerals were used to write rupee amounts, but there was a special notation for ānā. The sign ৹ always appears, and means sixteenths. Then. Prefixed to this is a numerator symbol, which goes ৴, ৵, ৶, ৷ for 1, 2, 3, 4. So for example, 3 ānā is written ৶৹, which means !!\frac3{16}!!. The larger fractions are made by adding the numerators, grouping by 4's:
except that three fours (৷৷৷) is too many, and is abbreviated by the intriguing NUMERATOR ONE LESS THAN THE DENOMINATOR sign ৸ when more than 11 ānā are being written. Historically, the ānā was divided into 20 gaṇḍā; the gaṇḍā amounts are written with standard (Benagli decimal) numerals instead of the special-purpose base-4 numerals just described. The gaṇḍā sign ৻ precedes the numeral, so 4 gaṇḍā (!!\frac15!! ānā) is wrtten as ৻৪. (The ৪ is not an 8, it is a four.) What if you want to write 17 rupees plus !!9\frac15!! ānā? That is 17 rupees plus 9 ānā plus 4 gaṇḍā. If I am reading this report correctly, you write it this way: ১৭৷৷৴৻৪ This breaks down into three parts as ১৭ ৷৷৴ ৻৪. The ১৭ is a 17, for 17 rupees; the ৷৷৴ means 9 ānā (the denominator ৹ is left implicit) and the ৻৪ means 4 gaṇḍā, as before. There is no separator between the rupees and the ānā. But there doesn't need to be, because different numerals are used! An analogous imaginary system in Latin script would be to write the amount as
where the ‘17’ means 17 rupees, the ‘dda’ means 4+4+1=9 ānā, and the ¢4 means 4 gaṇḍā. There is no trouble seeing where the ‘17’ ends and the ‘dda’ begins. Pandey says there was an even smaller unit, the kaṛi. It was worth ¼ of a gaṇḍā and was again written with the special base-4 numerals, but as if the gaṇḍā had been divided into 16. A complete amount might be written with decimal numerals for the rupees, base-4 numerals for the ānā, decimal numerals again for the gaṇḍā, and base-4 numerals again for the kaṛi. No separators are needed, because each section is written symbols that are different from the ones in the adjoining sections. [Other articles in category /math] permanent link Thu, 06 Aug 2020
Recommended reading: Matt Levine’s Money Stuff
Lately my favorite read has been Matt Levine’s Money Stuff articles from Bloomberg News. Bloomberg's web site requires a subscription but you can also get the Money Stuff articles as an occasional email. It arrives at most once per day. Almost every issue teaches me something interesting I didn't know, and almost every issue makes me laugh. Example of something interesting: a while back it was all over the news that oil prices were negative. Levine was there to explain what was really going on and why. Some people manage index funds. They are not trying to beat the market, they are trying to match the index. So they buy derivatives that give them the right to buy oil futures contracts at whatever the day's closing price is. But say they already own a bunch of oil contracts. If they can get the close-of-day price to dip, then their buy-at-the-end-of-the-day contracts will all be worth more because the counterparties have contracted to buy at the dip price. How can you get the price to dip by the end of the day? Easy, unload 20% of your contracts at a bizarre low price, to make the value of the other 80% spike… it makes my head swim. But there are weird second- and third-order effects too. Normally if you invest fifty million dollars in oil futures speculation, there is a worst-case: the price of oil goes to zero and you lose your fifty million dollars. But for these derivative futures, the price could in theory become negative, and for short time in April, it did:
One article I particularly remember discussed the kerfuffle a while back concerning whether Kelly Loeffler improperly traded stocks on classified coronavirus-related intelligence that she received in her capacity as a U.S. senator. I found Levine's argument persuasive:
He contrasted this case with that of Richard Burr, who, unlike Loeffler, remains under investigation. The discussion was factual and informative, unlike what you would get from, say, Twitter, or even Metafilter, where the response was mostly limited to variations on “string them up” and “eat the rich”. Money Stuff is also very funny. Today’s letter discusses a disclosure filed recently by Nikola Corporation:
A couple of recent articles that cracked me up discussed clueless day-traders pushing up the price of Hertz stock after Hertz had declared bankruptcy, and how Hertz diffidently attempted to get the SEC to approve a new stock issue to cater to these idiots. (The SEC said no.) One recurring theme in the newsletter is “Everything is Securities Fraud”. This week, Levine asks:
Of course you'd expect that the executives would be criminally charged, as they have been. But is there a cause for the company’s shareholders to sue? If you follow the newsletter, you know what the answer will be:
because Everything is Securities Fraud.
I recommend it. Levine also has a Twitter account but it is mostly just links to his newsletter articles. [ Addendum 20200821: Unfortunately, just a few days after I posted this, Matt Levin announced that his newletter would be on hiatus for a few months, as he would be on paternity leave. Sorry! ] [Other articles in category /ref] permanent link Wed, 05 Aug 2020
A maybe-interesting number trick?
I'm not sure if this is interesting, trivial, or both. You decide. Let's divide the numbers from 1 to 30 into the following six groups:
Choose any two rows. Chose a number from each row, and multiply them mod 31. (That is, multiply them, and if the product is 31 or larger, divide it by 31 and keep the remainder.) Regardless of which two numbers you chose, the result will always be in the same row. For example, any two numbers chosen from rows B and D will multiply to yield a number in row E. If both numbers are chosen from row F, their product will always appear in row A. [Other articles in category /math] permanent link Mon, 03 Aug 2020Gulliver's Travels (1726), Part III, chapter 2:
When I first told Katara about this, several years ago, instead of “the minds of these people are so taken up with intense speculations” I said they were obsessed with their phones. Now the phones themselves have become the flappers:
Our minds are not even taken up with intense speculations, but with Instagram. Dean Swift would no doubt be disgusted. [Other articles in category /book] permanent link Sat, 01 Aug 2020
How are finite fields constructed?
Here's another recent Math Stack Exchange answer I'm pleased with.
The only “reasonable” answer here is “get an undergraduate abstract algebra text and read the chapter on finite fields”. Because come on, you can't expect some random stranger to appear and write up a detailed but short explanation at your exact level of knowledge. But sometimes Internet Magic Lightning strikes and that's what you do get! And OP set themselves up to be struck by magic lightning, because you can't get a detailed but short explanation at your exact level of knowledge if you don't provide a detailed but short explanation of your exact level of knowledge — and this person did just that. They understand finite fields of prime order, but not how to construct the extension fields. No problem, I can explain that! I had special fun writing this answer because I just love constructing extensions of finite fields. (Previously: [1] [2]) For any given !!n!!, there is at most one field with !!n!! elements: only one, if !!n!! is a power of a prime number (!!2, 3, 2^2, 5, 7, 2^3, 3^2, 11, 13, \ldots!!) and none otherwise (!!6, 10, 12, 14\ldots!!). This field with !!n!! elements is written as !!\Bbb F_n!! or as !!GF(n)!!. Suppose we want to construct !!\Bbb F_n!! where !!n=p^k!!. When !!k=1!!, this is easy-peasy: take the !!n!! elements to be the integers !!0, 1, 2\ldots p-1!!, and the addition and multiplication are done modulo !!n!!. When !!k>1!! it is more interesting. One possible construction goes like this:
Now we will see an example: we will construct !!\Bbb F_{2^2}!!. Here !!k=2!! and !!p=2!!. The elements will be polynomials of degree at most 1, with coefficients in !!\Bbb F_2!!. There are four elements: !!0x+0, 0x+1, 1x+0, !! and !!1x+1!!. As usual we will write these as !!0, 1, x, x+1!!. This will not be misleading. Addition is straightforward: combine like terms, remembering that !!1+1=0!! because the coefficients are in !!\Bbb F_2!!: $$\begin{array}{c|cccc} + & 0 & 1 & x & x+1 \\ \hline 0 & 0 & 1 & x & x+1 \\ 1 & 1 & 0 & x+1 & x \\ x & x & x+1 & 0 & 1 \\ x+1 & x+1 & x & 1 & 0 \end{array} $$ The multiplication as always is more interesting. We need to find an irreducible polynomial !!P!!. It so happens that !!P=x^2+x+1!! is the only one that works. (If you didn't know this, you could find out easily: a reducible polynomial of degree 2 factors into two linear factors. So the reducible polynomials are !!x^2, x·(x+1) = x^2+x!!, and !!(x+1)^2 = x^2+2x+1 = x^2+1!!. That leaves only !!x^2+x+1!!.) To multiply two polynomials, we multiply them normally, then divide by !!x^2+x+1!! and keep the remainder. For example, what is !!(x+1)(x+1)!!? It's !!x^2+2x+1 = x^2 + 1!!. There is a theorem from elementary algebra (the “division theorem”) that we can find a unique quotient !!Q!! and remainder !!R!!, with the degree of !!R!! less than 2, such that !!PQ+R = x^2+1!!. In this case, !!Q=1, R=x!! works. (You should check this.) Since !!R=x!! this is our answer: !!(x+1)(x+1) = x!!. Let's try !!x·x = x^2!!. We want !!PQ+R = x^2!!, and it happens that !!Q=1, R=x+1!! works. So !!x·x = x+1!!. I strongly recommend that you calculate the multiplication table yourself. But here it is if you want to check: $$\begin{array}{c|cccc} · & 0 & 1 & x & x+1 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & x & x+1 \\ x & 0 & x & x+1 & 1 \\ x+1 & 0 & x+1 & 1 & x \end{array} $$ To calculate the unique field !!\Bbb F_{2^3}!! of order 8, you let the elements be the 8 second-degree polynomials !!0, 1, x, \ldots, x^2+x, x^2+x+1!! and instead of reducing by !!x^2+x+1!!, you reduce by !!x^3+x+1!!. (Not by !!x^3+x^2+x+1!!, because that factors as !!(x^2+1)(x+1)!!.) To calculate the unique field !!\Bbb F_{3^2}!! of order 27, you start with the 27 third-degree polynomials with coefficients in !!{0,1,2}!!, and you reduce by !!x^3+2x+1!! (I think). The special notation !!\Bbb F_p[x]!! means the ring of all polynomials with coefficients from !!\Bbb F_p!!. !!\langle P \rangle!! means the ring of all multiples of polynomial !!P!!. (A ring is a set with an addition, subtraction, and multiplication defined.) When we write !!\Bbb F_p[x] / \langle P\rangle!! we are constructing a thing called a “quotient” structure. This is a generalization of the process that turns the ordinary integers !!\Bbb Z!! into the modular-arithmetic integers we have been calling !!\Bbb F_p!!. To construct !!\Bbb F_p!!, we start with !!\Bbb Z!! and then agree that two elements of !!\Bbb Z!! will be considered equivalent if they differ by a multiple of !!p!!. To get !!\Bbb F_p[x] / \langle P \rangle!! we start with !!\Bbb F_p[x]!!, and then agree that elements of !!\Bbb F_p[x]!! will be considered equivalent if they differ by a multiple of !!P!!. The division theorem guarantees that of all the equivalent polynomials in a class, exactly one of them will have degree less than that of !!P!!, and that is the one we choose as a representative of its class and write into the multiplication table. This is what we are doing when we “divide by !!P!! and keep the remainder”. A particularly important example of this construction is !!\Bbb R[x] / \langle x^2 + 1\rangle!!. That is, we take the set of polynomials with real coefficients, but we consider two polynomials equivalent if they differ by a multiple of !!x^2 + 1!!. By the division theorem, each polynomial is then equivalent to some first-degree polynomial !!ax+b!!. Let's multiply $$(ax+b)(cx+d).$$ As usual we obtain $$acx^2 + (ad+bc)x + bd.$$ From this we can subtract !!ac(x^2 + 1)!! to obtain the equivalent first-degree polynomial $$(ad+bc) x + (bd-ac).$$ Now recall that in the complex numbers, !!(b+ai)(d + ci) = (bd-ac) + (ad+bc)i!!. We have just constructed the complex numbers,with the polynomial !!x!! playing the role of !!i!!. [ Note to self: maybe write a separate article about what makes this a good answer, and how it is structured. ] [Other articles in category /math/se] permanent link Fri, 31 Jul 2020
What does it mean to expand a function “in powers of x-1”?
A recent Math Stack Excahnge post was asked to expand the function !!e^{2x}!! in powers of !!(x-1)!! and was confused about what that meant, and what the point of it was. I wrote an answer I liked, which I am reproducing here. You asked:
which is a fair question. I didn't understand this either when I first learned it. But it's important for practical engineering reasons as well as for theoretical mathematical ones. Before we go on, let's see that your proposal is the wrong answer to this question, because it is the correct answer, but to a different question. You suggested: $$e^{2x}\approx1+2\left(x-1\right)+2\left(x-1\right)^2+\frac{4}{3}\left(x-1\right)^3$$ Taking !!x=1!! we get !!e^2 \approx 1!!, which is just wrong, since actually !!e^2\approx 7.39!!. As a comment pointed out, the series you have above is for !!e^{2(x-1)}!!. But we wanted a series that adds up to !!e^{2x}!!. As you know, the Maclaurin series works here: $$e^{2x} \approx 1+2x+2x^2+\frac{4}{3}x^3$$ so why don't we just use it? Let's try !!x=1!!. We get $$e^2\approx 1 + 2 + 2 + \frac43$$ This adds to !!6+\frac13!!, but the correct answer is actually around !!7.39!! as we saw before. That is not a very accurate approximation. Maybe we need more terms? Let's try ten: $$e^{2x} \approx 1+2x+2x^2+\frac{4}{3}x^3 + \ldots + \frac{8}{2835}x^9$$ If we do this we get !!7.3887!!, which isn't too far off. But it was a lot of work! And we find that as !!x!! gets farther away from zero, the series above gets less and less accurate. For example, take !!x=3.1!!, the formula with four terms gives us !!66.14!!, which is dead wrong. Even if we use ten terms, we get !!444.3!!, which is still way off. The right answer is actually !!492.7!!. What do we do about this? Just add more terms? That could be a lot of work and it might not get us where we need to go. (Some Maclaurin series just stop working at all too far from zero, and no amount of terms will make them work.) Instead we use a different technique. Expanding the Taylor series “around !!x=a!!” gets us a different series, one that works best when !!x!! is close to !!a!! instead of when !!x!! is close to zero. Your homework is to expand it around !!x=1!!, and I don't want to give away the answer, so I'll do a different example. We'll expand !!e^{2x}!! around !!x=3!!. The general formula is $$e^{2x} \approx \sum \frac{f^{(i)}(3)}{i!} (x-3)^i\tag{$\star$}\ \qquad \text{(when $x$ is close to $3$)}$$ The !!f^{(i)}(x)!! is the !!i!!'th derivative of !! e^{2x}!! , which is !!2^ie^{2x}!!, so the first few terms of the series above are: $$\begin{eqnarray} e^{2x} & \approx& e^6 + \frac{2e^6}1 (x-3) + \frac{4e^6}{2}(x-3)^2 + \frac{8e^6}{6}(x-3)^3\\ & = & e^6\left(1+ 2(x-3) + 2(x-3)^2 + \frac34(x-3)^3\right)\\ & & \qquad \text{(when $x$ is close to $3$)} \end{eqnarray} $$ The first thing to notice here is that when !!x!! is exactly !!3!!, this series is perfectly correct; we get !!e^6 = e^6!! exactly, even when we add up only the first term, and ignore the rest. That's a kind of useless answer because we already knew that !!e^6 = e^6!!. But that's not what this series is for. The whole point of this series is to tell us how different !!e^{2x}!! is from !!e^6!! when !!x!! is close to, but not equal to !!3!!. Let's see what it does at !!x=3.1!!. With only four terms we get $$\begin{eqnarray} e^{6.2} & \approx& e^6(1 + 2(0.1) + 2(0.1)^2 + \frac34(0.1)^3)\\ & = & e^6 \cdot 1.22075 \\ & \approx & 492.486 \end{eqnarray}$$ which is very close to the correct answer, which is !!492.7!!. And that's with only four terms. Even if we didn't know an exact value for !!e^6!!, we could find out that !!e^{6.2}!! is about !!22.075\%!! larger, with hardly any calculation. Why did this work so well? If you look at the expression !!(\star)!! you can see: The terms of the series all have factors of the form !!(x-3)^i!!. When !!x=3.1!!, these are !!(0.1)^i!!, which becomes very small very quickly as !!i!! increases. Because the later terms of the series are very small, they don't affect the final sum, and if we leave them out, we won't mess up the answer too much. So the series works well, producing accurate results from only a few terms, when !!x!! is close to !!3!!. But in the Maclaurin series, which is around !!x=0!!, those !!(x-3)^i!! terms are !!x^i!! terms intead, and when !!x=3.1!!, they are not small, they're very large! They get bigger as !!i!! increases, and very quickly. (The !! i! !! in the denominator wins, eventually, but that doesn't happen for many terms.) If we leave out these many large terms, we get the wrong results. The short answer to your question is:
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