问个数学问题

[amd]

sigma(n^2) = (2*n^3 + 3*n^2 + n)/6

sigma(n^2) = 1^2 + 2^2 + 3^2 + ... + n^2;

然后2边求导

sigma(2n) = (6*n^2 + 6*n + 1)/6

2*n(n+1)/2 = n^2 + n + 1/6

等式两边不等阿，数学丢了很多年了。不知道那里出错了。

哪个达人帮看看
谢谢。

 

[impression]

What's sigma? Is it a function? An unknown function?

最初由 amd 发布
sigma(n^2) = (2*n^3 + 3*n^2 + n)/6

然后2边求导

sigma(2n) = (6*n^2 + 6*n + 1)/6

2*n(n+1)/2 = n^2 + n + 1/6

等式两边不等阿，数学丢了很多年了。不知道那里出错了。

哪个达人帮看看
谢谢。

 

[一般化]

第2个SIGMA可不能那么求导啊。。。

最初由 amd 发布
sigma(n^2) = (2*n^3 + 3*n^2 + n)/6

sigma(n^2) = 1^2 + 2^2 + 3^2 + ... + n^2;

然后2边求导

sigma(2n) = (6*n^2 + 6*n + 1)/6

2*n(n+1)/2 = n^2 + n + 1/6

等式两边不等阿，数学丢了很多年了。不知道那里出错了。

哪个达人帮看看
谢谢。

 

[amd]

是不是说不能用求导和积分来算
1^2 + 2^2 + 3^2 + ... + n^2
的结果了？

 

[impression]

Hehe, it's interesting.
The summation is only for integers and integral is for continuous values.

The two result is similar with a little difference.
Sometime they use this technique as an approximation.

最初由 amd 发布
是不是说不能用求导和积分来算
1^2 + 2^2 + 3^2 + ... + n^2
的结果了？

 

[光之贱]

be careful of the continous and uniform

 

[泛蓝]

There's no surprise in this case.
Integers are not differentiable.
On the right side, it is a countinuous curve,
On the left, there is just some points on the curve.
Obviously the rate of change will not be the same unless
the curve is a straight line.

However, think about the fundamental of calculus,
you can see that as long as there's enough points to describe the curve,
the error will be considerably small.
So, it is widely used in engineering practice.
If you really want to learn how to prove whether there's enough
points,go look for books that talk about numerical method.
I am sure you will see it.

 

[amd]

谢谢了

 

[ACai]

最初由 amd 发布
是不是说不能用求导和积分来算
1^2 + 2^2 + 3^2 + ... + n^2
的结果了？

Differential must be continuous function and its differential function must exist.

For example f(x) = x^2; its f ' (x)= 2x. (x can be any number)
f(x)=1/x; when x = 0, its f ' (x) does not exist.