若已熟悉白努利周期函數之性質︰
。
果真知 ,白努利數唯一奇 ︰
既得白努利數之生成函數 且先探其奇偶性乎?
……
關係一現機鋒出 。原來這個白努利數唯一奇 , ,不假它求數自知 。遞迴關係無覓處
。恰恰此中得
☆
── 摘自《時間序列︰生成函數‧漸近展開︰白努利 □○《六》》
那麼一招一式之推導
能入『求和』 vs. 『求積』關聯之門也。
。
亦能曉
。
。
……
……
加之得
或
『取其便』之理矣。
如是者豈不會閱讀
證明
證明使用數學歸納法以及黎曼-斯蒂爾傑斯積分,下文中假設 的可微次數足夠大, 。
為了方便,將原式的各項用不同顏色表示:
的情形
容易算出
其中橙色的項通過分部積分可化為
假設 時原式成立
處理積分(藍色項)
將處理後的積分代入
得到想要的結果。
,焉不能得出公式
The formula
If and are natural numbers and is a complex or real valued continuous function for real numbers in the interval then the integral
can be approximated by the sum (or vice versa)
(see rectangle method). The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives evaluated at the end points of the interval, that is to say when and
Explicitly, for a natural number and a function that is times continuously differentiable in the interval we have
where is the th Bernoulli number (with ) and is an error term which is normally small for suitable values of and depends on and
The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for in which case we have[1][2]
or alternatively
耶?!