有趣的数学【01】:裂项法求和

in steempress •  22 days ago

我们先看一道数学题:
image
按照常规的做法,肯定是应该先通分,让这些数字的分母相同,然后分子相加。

这个通分可是不得了,很大:1×2×3×4×5×6×7×8×9×10×11,分子有好多,计算起来会很麻烦。

那么。我们现在动动脑筋,这样解题:
把第一项分解一下:
image

再分解第二项:
image

你可能会问:这样分解有什么用呢?仔细看看!!!

这样分解完以后,你会发现:第一项和第二项相加可以抵消一半的因子,-1/(2×3)和1/(2×3)可以抵消掉!!!

我们省略掉中间转换过程,最后算式变成了这样:
image

现在发现了吧,可以省略掉所有的中间项,只剩下第一子项和最后子项,最后算式很简单地就被算出来啦:
image


看到了没有?数学就是这样神奇。

实际上,上面的算法有一个标准的称呼,叫做:裂项法求和。

裂项法,这是分解与组合思想在数列求和中的具体应用。是将数列中的每项(通项)分解,然后重新组合,使之能消去一些项,最终达到求和的目的。


对于上述的算式,每一个加法因子可以表示成这样:
an=1/n(n+1)(n+2)

利用裂项法,将这个因子进行变换:
image

上面的算法,就是利用这个分解公式进行的。

此类变形的特点是将原数列每一项拆为两项之后,其中中间的大部分项都互相抵消了。只剩下有限的几项。

数列求和的常用方法有:
公式法、裂项相消法、错位相减法、倒序相加法等。

如果各位有兴趣的话,在今后的帖子中,我们可以逐个进行讨论。


参考:百度百科:裂项法


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