每日彙整: 2017-03-13

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︰破題

問題發現豈無因，

Compound interest

Jacob Bernoulli discovered this constant in 1683 by studying a question about compound interest:[5]

An account starts with ＄1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be ＄2.00. What happens if the interest is computed and credited more frequently during the year?

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial ＄1 is multiplied by 1.5 twice, yielding ＄1.00×1.52 = ＄2.25 at the end of the year. Compounding quarterly yields ＄1.00×1.254 = $2.4414…, and compounding monthly yields ＄ 1.00×(1+1/12)12 = ＄2.613035… If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be ＄1.00×(1 + 1/n)n.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields ＄2.692597…, while compounding daily (n = 365) yields  ＄2.714567…, just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach ＄2.7182818…. More generally, an account that starts at ＄1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. (Here R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05)

 

思維想法隨之來。

e (數學常數) 

e (mathematical constant)

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series[2]

{\displaystyle e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }

The constant can be characterized in many different ways. For example, e can be defined as the unique positive number a such that the graph of the function y = ax has unit slope at x = 0.[3] The function f(x) = ex is called the (natural) exponential function. The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a positive number k can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are alternative characterizations.

Sometimes called Euler’s number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler’s constant. The number e is also known as Napier’s constant, but Euler’s choice of the symbol e is said to have been retained in his honor.[4] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.[5]

The number e is of eminent importance in mathematics,[6] alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler’s identity. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995… (sequence A001113 in the OEIS).

 

古今歷史多少事？

History

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[5] However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli in 1683,[7][8] who attempted to find the value of the following expression (which is in fact e):

\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731.[9][10] Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,[11] and the first appearance of e in a publication was Euler’s Mechanica (1736).[12] While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.
convergence of the sequence (1+1/n)^n

 

Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Deep investigations by Dingle[1] reveal that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.

Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase “asymptotic series” usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics.[2] The error is then typically of the form  \sim\exp\left(-c / \epsilon\right) where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.

See asymptotic analysis, big O notation, and little o notation for the notation used in this article.

Formal definition

First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.

If φn is a sequence of continuous functions on some domain, and if L is a limit point of the domain, then the sequence constitutes an asymptotic scale if for every n,  \varphi_{n+1}(x) = o(\varphi_n(x)) \ (x \rightarrow L). (L may be taken to be infinity.) In other words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (in the limit  x \rightarrow L) than the preceding function.

If f is a continuous function on the domain of the asymptotic scale, then f has an asymptotic expansion of order N with respect to the scale as a formal series  \sum _{{n=0}}^{N}a_{n}\varphi _{{n}}(x) if

f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = O(\varphi_{N}(x)) \ (x \rightarrow L)

or

  f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = o(\varphi_{N-1}(x)) \ (x \rightarrow L).

If one or the other holds for all N, then we write

  f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \ (x \rightarrow L).

In contrast to a convergent series for  f, wherein the series converges for any fixed  x in the limit  N\rightarrow \infty , one can think of the asymptotic series as converging for fixed  N in the limit  x \rightarrow L ( L possibly infinite).

 

答案提出符號在？？

大O符號

大O符號（英語：Big O notation）是用於描述函數漸近行為的數學符號。更確切地說，它是用另一個（通常更簡單的）函數來描述一個函數數量級的漸近上界。在數學中，它一般用來刻畫被截斷的無窮級數尤其是漸近級數的剩餘項；在計算機科學中，它在分析算法複雜性的方面非常有用。

大O符號是由德國數論學家保羅·巴赫曼（Paul Bachmann）在其1892年的著作《解析數論》（Analytische Zahlentheorie）首先引入的。而這個記號則是在另一位德國數論學家艾德蒙·朗道（Edmund Landau）的著作中才推廣的，因此它有時又稱為朗道符號（Landau symbols）。代表「order of …」（……階）的大O，最初是一個大寫的希臘字母‘Ο‘（omicron），現今用的是大寫拉丁字母『O』，但從來不是阿拉伯數字『0』。

使用

這個符號有兩種形式上很接近但迥然不同的使用方法：無窮大漸近與無窮小漸近。然而這個區別只是在運用中的而不是原則上的——除了對函數自變量的一些不同的限定， 「大O」的形式定義在兩種情況下都是相同的。[來源請求]

無窮大漸近

大O符號在分析算法效率的時候非常有用。舉個例子，解決一個規模為  n的問題所花費的時間（或者所需步驟的數目）可以表示為： T(n)=4n^{2}-2n+2。當  n增大時，  n^{2}項將開始占主導地位，而其他各項可以被忽略。 舉例說明：當  n=500，  4n^{2}項是  2n項的1000倍大，因此在大多數場合下，省略後者對表達式的值的影響將是可以忽略不計的。

進一步看，如果我們與任一其他級的表達式比較，  n^{2}項的係數也是無關緊要的。例如：一個包含  n^{3}或  n^{2}項的表達式，即使 T(n)=1,000,000\cdot n^{2}，假定 U(n)=n^{3}，一旦  n增長到大於1,000,000，後者就會一直超越前者（T(1,000,000)=1,000,000^{3}=U(1,000,000)）。

這樣，大O符號就記下剩餘的部分，寫作：

T(n)\in \mathrm {O} (n^{2})

或

T(n)=\mathrm {O} (n^{2})

並且我們就說該算法具有  n^{2}階（平方階）的時間複雜度。

無窮小漸近

大O也可以用來描述數學函數估計中的誤差項。例如  e^{x}的泰勒展開：[來源請求]

e^{x}=1+x+{\frac {x^{2}}{2}}+{\hbox{O}}(x^{3})\qquad 當  x\to 0時

這表示，如果  x足夠接近於0，那麼誤差 e^{x}-\left(1+x+{\frac {x^{2}}{2}}\right)的絕對值小於  x^{3}的某一常數倍。

形式化定義

給定兩正值函數  f和  g,定義:

f(n)=\mathrm {O} (g(n)),條件為：存在正實數  c和  N,使得對於所有的  n\geq N,有  |f(n)|\leq |cg(n)|

上述的定義表明，當  n足夠大，大過一個特定的  N時，且存在一個正數  c,使得  |f|不大於  |cg|,則 f是  g的  \mathrm {O} 表示。  f和  g的關係可以理解為  f(n)是  g(n)的一個上界，也可以理解為  f最終至多增漲的速度與  g一樣快，但不會超過  g的增漲速度。