1 − 2 + 3 − 4 + …
1 − 2 + 3 − 4 + ⋯

context: zh_tw: 該等式的嚴謹解釋在很久以後才出現。自 1890 年起，恩納斯托・切薩羅、埃米爾・博雷爾與其他一些數學家就在研究有哪些定義良好的方法，可以給發散級數賦予廣義和 —— 其中包含了對尤拉結果的新解釋。這些求和法大部分可簡單地指定 1 − 2 + 3 − 4 + … 的 “和” 為 1⁄4。切薩羅求和是少數幾種不能計算出 1 − 2 + 3 − 4 + … 之和的方法，因為此級數求和需要某個略強的方法 —— 譬如阿貝耳求和。
en: A rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Cesàro, Émile Borel and others investigated well-defined methods to assign generalized sums to divergent series—including new interpretations of Euler's attempts. Many of these summability methods easily assign to 1 − 2 + 3 − 4 + ... a "value" of 1/4. Cesàro summation is one of the few methods that do not sum 1 − 2 + 3 − 4 + ..., so the series is an example where a slightly stronger method, such as Abel summation, is required.
qas: question: zh_tw: 什麼時候 ernesto cesaro 研究了將 sms 分配到不同系列的方法？
en: When did Ernesto Cesaro study methods to assign sms to divergent series?
answers: zh_tw: text: 1890 年
en: text: 1890

context: zh_tw: 級數 1 − 2 + 3 − 4 + … 與格蘭迪級數 1 − 1 + 1 − 1 + … 有緊密的聯絡。尤拉將這兩個級數當作 1 − 2n + 3n − 4n + … 的特例（其中 n 為任意自然數），這個級數既直接擴充套件了他在巴塞爾問題上所做的工作，同時也引出了我們現在所知的狄利克雷 η 函式和黎曼 ζ 函式。
en: The series 1 − 2 + 3 − 4 + ... is closely related to Grandi's series 1 − 1 + 1 − 1 + .... Euler treated these two as special cases of 1 − 2n + 3n − 4n + ... for arbitrary n, a line of research extending his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function.
qas: question: zh_tw: 從巴塞爾問題而來的兩個函式是什麼？
en: what are the two functions that came from work on the Basel problem?
answers: zh_tw: text: 狄利克雷 η 函式和黎曼 ζ 函式
en: text: the Dirichlet eta function and the Riemann zeta function

context: zh_tw: 切薩羅求和有兩種有名的廣義化：讓這些在概念上更簡單的是 (H, n) 法的序列，其中 n 為自然數。(H, 1) 和為切薩羅求和，更高的方法則重複平均值的計算。在上文中，偶數項平均值趨近於 1⁄2，奇數項平均值則全部等於 0，所以平均值的平均值趨近於 0 與 1⁄2 的平均數，即 1⁄4。因此，1 − 2 + 3 − 4 + … 是 (H, 2)- 可求和，其值為 1⁄4。
en: There are two well-known generalizations of Cesàro summation: the conceptually simpler of these is the sequence of (H, n) methods for natural numbers n. The (H, 1) sum is Cesàro summation, and higher methods repeat the computation of means. Above, the even means converge to ​1⁄2, while the odd means are all equal to 0, so the means of the means converge to the average of 0 and ​1⁄2, namely ​1⁄4. So 1 − 2 + 3 − 4 + ... is (H, 2) summable to ​1⁄4.
qas: question: zh_tw: 在切薩羅求和中，偶數平均和奇數平均的平均值是多少？
en: In the Cesaro summation, what do the averages of the even means and the odd means converge to?
answers: zh_tw: text: 1⁄4
en: text: ​1⁄4

context: zh_tw: 符號 “H” 代表奧圖・赫爾德。1882 年，他第一次證明了被現在數學家們所看作的在阿貝耳求和與 (H, n) 求和之間的關係；-1 + 2 − 3 + 4 − … 是他給的第一個例子。1⁄4 是 1 − 2 + 3 − 4 + … 的 (H, 2) 和這個事實也保證了它是阿貝耳和；這些都將在下文直接予以證明。
en: The "H" stands for Otto Hölder, who first proved in 1882 what mathematicians now think of as the connection between Abel summation and (H, n) summation; 1 − 2 + 3 − 4 + ... was his first example. The fact that ​1⁄4 is the (H, 2) sum of 1 − 2 + 3 − 4 + ... guarantees that it is the Abel sum as well; this will also be proved directly below.
qas: question: zh_tw: 奧圖何時在數學領域證明了一些東西？
en: When did Otto prove something in the field of mathematics?
answers: zh_tw: text: 1882
en: text: 1882

context: zh_tw: 另外一個常用的切薩羅求和的廣義化，是 (C, n) 法的序列。已經證明了 (C, n) 求和與 (H, n) 求和均能給出相同的結果，但是它們卻有不同的歷史背景。在 1887 年，切薩羅已經接近於陳述出 (C, n) 求和的定義了，但是他只給出了少量的例子。特別的，他在計算 1 − 2 + 3 − 4 + … 為 1⁄4 時所採用的方法可能是 (C, n) 的另一種描述，但是在當時並沒有對其進行證明。他在 1890 年正式定義了 (C, n) 法，以陳述他的定理：一個 (C, n)- 可求和級數與一個 (C, m)- 可求和級數的柯西乘積是 (C, m + n + 1)- 可求和。
en: The other commonly formulated generalization of Cesàro summation is the sequence of (C, n) methods. It has been proven that (C, n) summation and (H, n) summation always give the same results, but they have different historical backgrounds. In 1887, Cesàro came close to stating the definition of (C, n) summation, but he gave only a few examples. In particular, he summed 1 − 2 + 3 − 4 + ..., to ​1⁄4 by a method that may be rephrased as (C, n) but was not justified as such at the time. He formally defined the (C, n) methods in 1890 in order to state his theorem that the Cauchy product of a (C, n)-summable series and a (C, m)-summable series is (C, m + n + 1)-summable.
qas: question: zh_tw: 卡西奧定義了什麼方法？
en: what methods did Cesaro define?
answers: zh_tw: text: 他在 1890 年正式定義了 (C, n) 法，以陳述他的定理：一個 (C,
en: text: the (C, n) methods

question: zh_tw: (H, n) 求和與什麼求和均能給出相同的結果？
en: What other summation sequence always gives teh same result as (H,n)?
answers: zh_tw: text: (C, n)
en: text: (C, n)

context: zh_tw: 尤拉曾幾次提議將 “和” 這個詞廣義化。在 1 − 2 + 3 − 4 + … 的情況下，他的設想與現在所知的阿貝耳求和相似：
en: Euler proposed a generalization of the word "sum" several times. In the case of 1 − 2 + 3 − 4 + ..., his ideas are similar to what is now known as Abel summation:
qas: question: zh_tw: 尤拉的設想和什麼相似？
en: What are Eulers idea similar to?
answers: zh_tw: text: 阿貝耳求和
en: text: Abel summation

context: zh_tw: 可以對右邊作泰勒展開，或使用正規的多項式長除。從左方開始，可採用上文的一般啟發式，並嘗試乘以兩次 (1+x)，或對幾何級數 1 − x + x2 − … 求平方。尤拉似乎也提出可以對後者級數的每項求導。以現代的觀點看，級數 1 − 2x + 3x2 − 4x3 + … 並沒有定義一個在 x = 1 時的函式，因此不能簡單地把值代入到其相應的表示式。不過由於此級數在 | x| < 1 時定義了一個函式，所以仍可取 x 趨近於 1 時的極限，而這就是阿貝耳和的定義：
en: One can take the Taylor expansion of the right-hand side, or apply the formal long division process for polynomials. Starting from the left-hand side, one can follow the general heuristics above and try multiplying by (1 + x) twice or squaring the geometric series 1 − x + x2 − .... Euler also seems to suggest differentiating the latter series term by term.In the modern view, the series 1 − 2x + 3x2 − 4x3 + ... does not define a function at x = 1, so that value cannot simply be substituted into the resulting expression. Since the function is defined for all |x| < 1, one can still take the limit as x approaches 1, and this is the definition of the Abel sum:
qas: question: zh_tw: 除了展開泰勒，解決表示式的另一個方法是什麼？
en: What is another way besides the Taylor expansion to solve the expression?
answers: zh_tw: text: 正規的多項式長除
en: text: formal long division process

context: zh_tw: 下一步需要 1, 2, 3, 4, … 的前向差分序列；這恰好是 1, 1, 1, 1, …。將該序列的首項標示為 Δa0。尤拉變換也基於差分的差分，以及更高的迭代，但是在 1, 1, 1, 1, … 各項之間的前向差分均為 0。1 − 2 + 3 − 4 + … 的尤拉變換便可定義為：
en: Next one needs the sequence of forward differences among 1, 2, 3, 4, ...; this is just 1, 1, 1, 1, .... The first element of this sequence is labeled Δa0. The Euler transform also depends on differences of differences, and higher iterations, but all the forward differences among 1, 1, 1, 1, ... are 0. The Euler transform of 1 − 2 + 3 − 4 + ... is then defined as
qas: question: zh_tw: 尤拉變換基於什麼？
en: What does Euler transform depend on?
answers: zh_tw: text: 差分的差分，以及更高的迭代，但是在 1, 1, 1, 1, … 各項之間的前向差分均為 0
en: text: differences of differences, and higher iterations, but all the forward differences among 1, 1, 1, 1, ... are 0

context: zh_tw: 該結果推廣了阿貝耳求和，當取 φ(x) = exp (−x) 時可得到先前的等式。此一般陳述可透過將關於 m 的級數中的項配對，並將表示式變換為黎曼積分的形式予以證明。在後一步中，對 1 − 1 + 1 − 1 + … 的相應證明運用了中值定理，但在這裡需要泰勒公式中更強的拉格朗日形式。
en: This result generalizes Abel summation, which is recovered by letting φ(x) = exp(−x). The general statement can be proved by pairing up the terms in the series over m and converting the expression into a Riemann integral. For the latter step, the corresponding proof for 1 − 1 + 1 − 1 + ... applies the mean value theorem, but here one needs the stronger Lagrange form of Taylor's theorem.
qas: question: zh_tw: 如何復原阿貝耳求和？
en: How is the Abel summation recovered?
answers: zh_tw: text: 當取 φ(x) = exp (−x)
en: text: by letting φ(x) = exp(−x)

context: zh_tw: 0 = 1 − 2n + 3n − 4n + etc. 其中，n 為正數。這是一個笑料，朋友。” 切薩羅的老師歐仁・查理・卡塔蘭也輕視發散級數。在卡塔蘭的影響下，切薩羅早期提出 1 − 2n + 3n − 4n + … 的 “習用式” 是 “荒謬的等式”；而在 1883 年，切薩羅表明了當時的一個典型看法：這些公式是錯的，不過在某些場合在形式上是有用的。最後，在他 1890 年的書《Sur la multiplication des séries》中，切薩羅從定義開始採用了一個現代的做法。此級數在 n 為非整數值的情況亦有所研究；這產生了狄利克雷 η 函式。尤拉研究 1 − 2 + 3 − 4 + … 相關級數的部分動機是 η 函式的函式方程，這直接導向了黎曼 ζ 函式的函式方程。尤拉在正偶數（包括在巴塞爾問題中）時找到這些函式值的建樹已讓他聞名，他也試圖找到正奇數（包括在阿培裡常數中）時的值，但這個問題直到今天仍是難以解決的。η 函式透過尤拉的方法解決會比較簡單，因為它的狄利克雷級數是處處阿貝耳可求和；而 ζ 函式的狄利克雷級數則非常難以對發散的部分求和。例如，1 − 2 + 3 − 4 + … 在 η 函式中的相似級數是非交錯級數 1 + 2 + 3 + 4 + …，該級數在現代物理學上有很深的應用，不過需要非常強的方法才能求和。
en: Cesàro's teacher, Eugène Charles Catalan, also disparaged divergent series. Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for 1 − 2n + 3n − 4n + ... as "absurd equalities", and in 1883 Cesàro expressed a typical view of the time that the formulas were false but still somehow formally useful. Finally, in his 1890 Sur la multiplication des séries, Cesàro took a modern approach starting from definitions.The series are also studied for non-integer values of n; these make up the Dirichlet eta function. Part of Euler's motivation for studying series related to 1 − 2 + 3 − 4 + ... was the functional equation of the eta function, which leads directly to the functional equation of the Riemann zeta function. Euler had already become famous for finding the values of these functions at positive even integers (including the Basel problem), and he was attempting to find the values at the positive odd integers (including Apéry's constant) as well, a problem that remains elusive today. The eta function in particular is easier to deal with by Euler's methods because its Dirichlet series is Abel summable everywhere; the zeta function's Dirichlet series is much harder to sum where it diverges. For example, the counterpart of 1 − 2 + 3 − 4 + ... in the zeta function is the non-alternating series 1 + 2 + 3 + 4 + ..., which has deep applications in modern physics but requires much stronger methods to sum.
qas: question: zh_tw: 切薩羅持典型的看法時是什麼時候？
en: When did Cesàro hold the standard view at the time?
answers: zh_tw: text: 1883
en: text: 1883