此帖非常重要，所以我寫了中英對照版本。
This post is really important so we have Chinese and English version.
我是專門研究代數的數學家。
當我第一次用差分算子求出方冪和的時候，我感到驚訝。
方冪和有公式，那個公式有伯努利數。
不過從朱世杰恆等式可以看到這類求和可以有更簡單的表達式。
其實我們只要用牛頓級數將多項式寫成組合數，就可以求出方冪和。
同樣地，當我們想求出差比數列的和時，
可能打算從等比數列遞推上去，我們根本沒有公式。
我後悔學了這麼久才知道有差分算子。
我只是想大家將來能夠更早知道這件事。
這是一個被定義成f(x+1)-f(x)的算子的故事。
我認為在學微分之前就應該先學差分。
可惜我學完微分之後還沒有聽說過有這種東西。
I am a mathematician interested in algebra though I'm just working as an accountant in Hong Kong.
I am surprised that Finite Difference has become an excellent tool on power sum.
While Faulhaber's formula is well-known as a general formula to this kind of problem, we found that the Hockey-stick identity is able to deal with some kind of power sum with a very simple formula.
We can rewrite a polynomial to binomial coefficient with Newton's series, so the power sum can be expressed by a much clear form with Finite Difference.
Again, when we're going to find the sum of arithmetic–geometric sequence, we're just expected to derive it from the sum of geometric sequence.
We don't even know a general formula for this problem.
I regret that I learn the Finite Difference so late.
I just hope that everyone would have the chance to learn this earlier.
Here's a story with something defined as f(x+1)-f(x).
It's a Linear operator which is something we should study before differentiation.
And I still haven't heard of it after studying differentiation.
詳情可見條目求和符號及維基教科書代數/本書課文/求和
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