每日彙整: 2017-03-29

樹莓派、樹莓派之學習、樹莓派之教育

時間序列︰生成函數‧漸近展開︰白努利 □○《九上》

略讀 John D. Blanton 先生之

《Foundations Of Differential Calculus: 1st (first) Edition.》

後，發現不巧只翻譯了歐拉

《Institutiones calculi differentialis》

巨著的第一部份。所以在此藉著莫里斯·克萊因 

Morris Kline

Morris Kline (May 1, 1908 – June 10, 1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.

教授一九八三在數學雜誌上發表之

《Euler and Infinte Series》

文章說說歐拉『形式操作』觀點的了。同時趁機介紹讀者認識這位『數學教育』知名的批評者 Kline 先生︰

Critique of mathematics education

Morris Kline was a protagonist in the curriculum reform in mathematics education that occurred in the second half of the twentieth century, a period including the programs of the new math. An article by Kline in 1956 in The Mathematics Teacher, the main journal of the National Council of Teachers of Mathematics, was titled “Mathematical texts and teachers: a tirade“. Calling out teachers blaming students for failures, he wrote “There is a student problem, but there are also three other factors which are responsible for the present state of mathematical learning, namely, the curricula, the texts, and the teachers.” The tirade touched a nerve, and changes started to happen. But then Kline switched to being a critic of some of the changes. In 1958 he wrote “Ancients versus moderns: a new battle of the books“. The article was accompanied with a rebuttal by Albert E. Meder Jr. of Rutgers University.[2] He says, “I find objectionable: first, vague generalizations, entirely undocumented, concerning views held by ‘modernists’, and second, the inferences drawn from what has not been said by the ‘modernists’.” By 1966 Kline proposed an eight-page high school plan.[3] The rebuttal for this article was by James H. Zant; it asserted that Kline had “a general lack of knowledge of what was going on in schools with reference to textbooks, teaching, and curriculum.” Zant criticized Kline’s writing for “vagueness, distortion of facts, undocumented statements and overgeneralization.”

In 1966[4] and 1970[5] Kline issued two further criticisms. In 1973 St. Martin’s Press contributed to the dialogue by publishing Kline’s critique, Why Johnny Can’t Add: the Failure of the New Math. Its opening chapter is a parody of instruction as students’ intuitions are challenged by the new jargon. The book recapitulates the debates from Mathematics Teacher, with Kline conceding some progress: He cites Howard Fehr of Columbia University who sought to unify the subject through its general concepts, sets, operations, mappings, relations, and structure in the Secondary School Mathematics Curriculum Improvement Study.

In 1977 Kline turned to undergraduate university education; he took on the academic mathematics establishment with his Why the Professor Can’t Teach: the dilemma of university education. Kline argues that onus to conduct research misdirects the scholarly method that characterizes good teaching. He lauds scholarship as expressed by expository writing or reviews of original work of others. For scholarship he expects critical attitudes to topics, materials and methods. Among the rebuttals are those by D.T. Finkbeiner, Harry Pollard, and Peter Hilton.[6] Pollard conceded, “The society in which learning is admired and pursued for its own sake has disappeared.” The Hilton review was more direct: Kline has “placed in the hand of enemies…[a] weapon”. Having started in 1956 as an agitator for change in mathematics education, he became a critic of some trends. Skilled expositor that he was, editors frequently felt his expressions were best tempered with rebuttal.

In considering what motivated Morris Kline to protest, consider Professor Meder’s opinion:[7]I am wondering whether in point of fact, Professor Kline really likes mathematics […] I think that he is at heart a physicist, or perhaps a ‘natural philosopher’, not a mathematician, and that the reason he does not like the proposals for orienting the secondary school college preparatory mathematics curriculum to the diverse needs of the twentieth century by making use of some concepts developed in mathematics in the last hundred years or so is not that this is bad mathematics, but that it minimizes the importance of physics.

It might appear so, as Kline recalls E. H. Moore’s recommendation to combine science and mathematics at the high school level.[8] But closer reading shows Kline calling mathematics a “part of man’s efforts to understand and master his world“, and he sees that role in a broad spectrum of sciences.

 

為著方便讀者閱讀理解克萊因教授文章，勉力註解梳理一番。

數學上如何考慮

S = 1 - 1 + 1 - 1 + \cdots

計算所引起的爭議呢？

(1) \ S = 0 = (1 - 1) + (1 - 1) + \cdots

(2) \ S = 1 = 1 - (1 - 1) - (1 - 1) - \cdots

有限項『代數表達式』建立之法則 ── 比方說『加括號』───，能否擴張於無窮 \infty 耶？就今日所知『項次安排』有條件也︰

Rearrangements

For any series, we can create a new series by rearranging the order of summation. A series is unconditionally convergent if any rearrangement creates a series with the same convergence as the original series. Absolutely convergent series are unconditionally convergent. But the Riemann series theorem states that conditionally convergent series can be rearranged to create arbitrary convergence.[1] The general principle is that addition of infinite sums is only commutative for absolutely convergent series.

For example, one false proof that 1=0 exploits the failure of associativity for infinite sums.

As another example, we know that

  \ln(2)=\sum _{{n=1}}^{\infty }{\frac {(-1)^{{n+1}}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots .

※

墨卡托級數

在數學內，墨卡托級數(Mercator series)或者牛頓-墨卡托級數(Newton–Mercator series)是一個自然對數的泰勒級數：

{\displaystyle \ln(1+x)\;=\;x\,-\,{\frac {x^{2}}{2}}\,+\,{\frac {x^{3}}{3}}\,-\,{\frac {x^{4}}{4}}\,+\,\cdots .}

使用大寫sigma表示則為

{\displaystyle \ln(1+x)\;=\;\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}.}

當 −1 < x ≤ 1時，此級數收斂於自然對數(加了1)。

───

But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for  {\frac {1}{2}}\ln(2):

{\begin{aligned}&{}\quad \left(1-{\frac {1}{2}}\right)-{\frac {1}{4}}+\left({\frac {1}{3}}-{\frac {1}{6}}\right)-{\frac {1}{8}}+\left({\frac {1}{5}}-{\frac {1}{10}}\right)-{\frac {1}{12}}+\cdots \\[8pt]&={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{10}}-{\frac {1}{12}}+\cdots \\[8pt]&={\frac {1}{2}}\left(1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots \right)={\frac {1}{2}}\ln(2).\end{aligned}}

 

若問 (1), \ (2) 都『加無窮括號』與

(3) \ S = 1 - (1 - 1 + 1 - \cdots) = 1 - S

加括號法相同嗎？？恐有懸念乎！！

現今數學重視歐拉『級數變換』想法︰

Ordinary generating function

The transform connects the generating functions associated with the series. For the ordinary generating function, let

  f(x)=\sum_{n=0}^\infty a_n x^n

and

  g(x)=\sum_{n=0}^\infty s_n x^n

then

  {\displaystyle g(x)=(Tf)(x)={\frac {1}{1-x}}f\left(-{\frac {x}{1-x}}\right).}

Euler transform

The relationship between the ordinary generating functions is sometimes called the Euler transform. It commonly makes its appearance in one of two different ways. In one form, it is used to accelerate the convergence of an alternating series. That is, one has the identity

  \sum_{n=0}^\infty (-1)^n a_n = \sum_{n=0}^\infty (-1)^n \frac {\Delta^n a_0} {2^{n+1}}

which is obtained by substituting x=1/2 into the last formula above. The terms on the right hand side typically become much smaller, much more rapidly, thus allowing rapid numerical summation.

 

推廣了歐拉『可加性』概念︰

Euler summation

In the mathematics of convergent and divergent series, Euler summation is a summability method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series ∑an, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series.

Euler summation can be generalized into a family of methods denoted (E, q), where q ≥ 0. The (E, 1) sum is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation; for q > 0 they are incomparable with Abel summation.

Definition

For some value y we may define the Euler sum (if it converges for that value of y) corresponding to a particular formal summation as:

{\displaystyle _{E_{y}}\,\sum _{j=0}^{\infty }a_{j}:=\sum _{i=0}^{\infty }{\frac {1}{(1+y)^{i+1}}}\sum _{j=0}^{i}{\binom {i}{j}}y^{j+1}a_{j}.}

If the formal sum actually converges, an Euler sum will equal it. But Euler summation is particularly used to accelerate the convergence of alternating series and sometimes it can give a useful meaning to divergent sums.

To justify the approach notice that for interchanged sum, Euler’s summation reduces to the initial series, because

  {\displaystyle y^{j+1}\sum _{i=j}^{\infty }{\binom {i}{j}}{\frac {1}{(1+y)^{i+1}}}=1.}

※

因為

\sum \limits_{n=k}^{\infty} \left( \begin{array}{ccc} n \\ k \end{array} \right) y^n = \frac{y^k}{{(1-y)}^{k+1}}

假設 z= \frac{1}{1+y} ，於是 y = \frac{1-z}{z}。

因此

y^{j+1} \sum \limits_{i=j}^{\infty} \left( \begin{array}{ccc} i \\ j \end{array} \right) \frac{1}{{(1+y)}^{i+1}}

改寫為

= { \left( \frac{1-z}{z} \right) }^ {j+1} \sum \limits_{i=j}^{\infty} \left( \begin{array}{ccc} i \\ j \end{array} \right) z^{j+1}

= { \left( \frac{1-z}{z} \right) }^ {j+1} \cdot z \cdot \frac{z^j}{{(1-z)}^{j+1}}

=1 。

Generating functions

Ordinary generating functions

For a fixed n, the ordinary generating function of the sequence  {n \choose 0},\;{n \choose 1},\;{n \choose 2},\;\ldots is:

  {\displaystyle \sum _{k=0}^{\infty }{n \choose k}x^{k}=(1+x)^{n}.}

For a fixed k, the ordinary generating function of the sequence  {0 \choose k},\;{1 \choose k},\;{2 \choose k},\;\ldots is:

  \sum _{n=k}^{\infty }{n \choose k}y^{n}={\frac {y^{k}}{(1-y)^{k+1}}}.

The bivariate generating function of the binomial coefficients is:

  {\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{n}{n \choose k}x^{k}y^{n}={\frac {1}{1-y-xy}}.}

Another bivariate generating function of the binomial coefficients, which is symmetric, is:

{\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{n+k \choose k}x^{k}y^{n}={\frac {1}{1-x-y}}.}

Exponential generating function

A symmetric exponential bivariate generating function of the binomial coefficients is:

{\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{n+k \choose k}{\frac {x^{k}y^{n}}{(n+k)!}}=e^{x+y}.}

This method itself cannot be improved by iterated application, as

  _{{E_{{y_{1}}}}}{}_{{E_{{y_{2}}}}}\sum =\,_{{E_{{{\frac {y_{1}y_{2}}{1+y_{1}+y_{2}}}}}}}\sum .

 

終究不要忘記尚有許多種未必相容之『求和法』觀點哩︰

發散級數

發散級數指（按柯西意義下）不收斂的級數。如級數  1 + 2 + 3 + 4 + \cdots和  1 - 1 + 1 - 1 + \cdots

但在實際的數學研究及物理等其它學科的應用中，經常需對發散級數進行運算，於是數學家們便給發散級數定義各種不同的「和」，如切薩羅和、阿貝爾和、歐拉和等，使對收斂級數求得的這些和仍然不變，而對某些發散級數，這種和仍然存在。

各種求和法

切薩羅和

主條目：切薩羅求和

對於級數  \sum_{n=1}^{\infty}a_n,令  s_n = a_1 + \cdots + a_n為它的部分和，而  t_n = \frac{s_1 + \cdots + s_n}{n}。如果  t_n \rightarrow s，則稱這個級數的切薩羅和為  s。

阿貝爾和

如果冪級數 \sum_{n=0}^{\infty}a_n x^n在  |x|<1收斂，並且 \lim_{x \rightarrow 1^- }\sum_{n=0}^{\infty}a_n x^n = s，則稱級數   \sum_{n=0}^{\infty}a_n的阿貝爾和為s。

拉馬努金求和約定

如果指數母函數  \sum_{n=1}^{\infty}a_n e^{-nz}的收斂區域非空，且它可以解析延拓為複平面上的亞純函數，它的洛朗級數的零次係數就等於級數  \sum_{n=1}^{\infty}a_n的拉馬努金和[1]。

例如，我們有以下級數的拉馬努金和：

  1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}(\Re).
  1 + 1 + 1 + 1 + \cdots = -\frac{1}{2}(\Re).
  1 - 1 + 1 - 1 + \cdots = \frac{1}{2}(\Re).

 

那麼歐拉認為 \infty < -1 ，無限大\infty 像零 0 一般分隔著『正、負』數，有理耶！！無理哉？？