一题简单的数学题

streamleaf

考考大家一题看似简单的数学题：

1 - 1 + 1 - 1 + 1 - 1 + ...... (到 infinity) = ?

答对没奖  

shitleg

1?

对吗?

qiqimon5566

0

streamleaf

不是 1，也不是 0
答案很神奇.....

jimmy8349

答案是∞

streamleaf

回复 5# jimmy8349

也不是

Pax

计算机爆表。。。。

m叶子m

答案是        ，对吧？






P.S.楼主玩神秘，我也玩

doktorkoo

答案是 NOt Responding 呵呵

丘比特

答案可能是 infinity  , 或者 - infinity ?

streamleaf

也不是 +/- infinity......

答案是。。。。  0.5

至于怎么算到 0.5，大家可以试试看

kc_kc

會算到0.5,是不是計算機壞了啊?

0secret0

好奇下
这题infinity还能有答案？

baba0000

说infinity的朋友要看清楚题目。说0或1的朋友的答案虽然不完全对，但也应该看出这series在不同的情况下，会产生两个答案。但是如果我们这样看：

假设这series为x，则

x = 1 - 1 + 1 - 1 + 1 - 1 + ......

1 - x = 1 - (1 - 1 + 1 - 1 + 1 - 1 + ......）

1 - x = 1 - 1 + 1 - 1 + 1 - 1 + ......
在这里，有没有发现等式的右边其实就是一开始假设的x。所以，

1 - x = x

x = 1/2 得证。

streamleaf

回复 14# baba0000

谢谢解说
无论是在 series 等于 0 或 1 时停止计算都不正确，因为 infinity 原本就是无限大。

貌似不可能的东西却能那么容易地用算术解释真的很神奇。

更神奇的是在现实中也有例子（传热学）来验证它的正确性。

madara_tachi

朋友，你错了。 答案不是 0.5

That (infinite) series diverges by bounded oscillation,since the sequence of partial sums is (1, 0, 1, 0, 1,0, …..). That is, the sequence of partial sums does not converge.

You let x equals to that infinite series, however you can only do this if that infinite series actually does converge, which you assume it is. (While it isn’t) I know you probably use this method for (somewhat) similar problems before, but you can do that because the series in the questions that you encountered before do converge to a value, unlike this series.

streamleaf

S = 1 - 1 + 1 - 1 + ... 是一个 Grandi's Divergent Series。

如果把这 series 当成 Divergent Geometric Series, 那我们可以用楼上的方式计算出 S = 1/2 [Devlin, 1994] (即使它 diverges)


或者我们可以用 Cesaro Summation。 Cesaro [1859-1906] 理论的其中一点是如果这个 series 不 converge, 但也不 diverge to an infinite value，那么就有可能有一个 finite summation value (1,0,1,0,1,0,.... 虽不converge，但也不 diverge to infinity)

Cesaro 的公式是， Cesaro sum =

s_k: partial sum of the series.

试一试 Cesaro 的理论，这个 sequence 就是 1/1, 1/2, 2/3, 2/4, 3/5, 3/6, 4/7, 4/8, 5/9 ..... (approaches 1/2 as n approaches infinity).


当然也有数学家觉得不是 0 就是 1.

我不是数学家，所以只选对我有实际用途的答案 - 0.5


如有读数学系的朋友，也请帮忙指点迷津

madara_tachi

S = 1 - 1 + 1 - 1 + ... 是一个 Grandi's Divergent Series。

如果把这 series 当成 Divergent Geometr ...
streamleaf 发表于 17-4-2010 10:08 PM

What Cesaro summation does is tracking the average of the partial sums, which in this case is 0.5 It is NOT the exact value of this infinite series, because this infinite series does not even converge. I didn't say that this series diverges to infinity, I just said that it diverges. When you say that it diverges, it does not necessarily diverge to infinity, it can diverge by bounded oscillation (this case).

中文

madara_tachi

S = 1 - 1 + 1 - 1 + ... 是一个 Grandi's Divergent Series。

如果把这 series 当成 Divergent Geometr ...
streamleaf 发表于 17-4-2010 10:08 PM

What Cesaro summation does is tracking the average of the partial sums, which in this case is 0.5 It is NOT the exact value of this infinite series, because this infinite series does not even converge. I didn't say that this series diverges to infinity, I just said that it diverges. When you say that it diverges, it does not necessarily diverge to infinity, it can diverge by bounded oscillation (this case).

中文

madara_tachi

S = 1 - 1 + 1 - 1 + ... 是一个 Grandi's Divergent Series。

如果把这 series 当成 Divergent Geometr ...
streamleaf 发表于 17-4-2010 10:08 PM

What Cesaro summation does is tracking the average of the partial sums, which in this case is 0.5 It is NOT the exact value of this infinite series, because this infinite series does not even converge. I didn't say that this series diverges to infinity, I just said that it diverges. When you say that it diverges, it does not necessarily diverge to infinity, it can diverge by bounded oscillation (this case).

中文