有一分数序列求前20项和答案

一、序列的定义和性质

这个分数序列的通项公式为 $a_n = \dfrac{n}{n+1}$，其中 $n$ 是正整数。其前 $20$ 项分别为：

$$\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{5}{6},\dfrac{6}{7},\dfrac{7}{8},\dfrac{8}{9},\dfrac{9}{10},\dfrac{10}{11},\dfrac{11}{12},\dfrac{12}{13},\dfrac{13}{14},\dfrac{14}{15},\dfrac{15}{16},\dfrac{16}{17},\dfrac{17}{18},\dfrac{18}{19},\dfrac{19}{20},\dfrac{20}{21}$$

我们可以看到，这个分数序列的每一项都比前一项稍大一点，但是增长速度逐渐减缓。实际上，当 $n$ 很大时，这个分数序列的每一项都非常接近于 $1$。

二、前 $20$ 项的和

要求前 $20$ 项的和，我们可以直接计算：

$$\begin{aligned} \sum_{n=1}^{20} a_n &= \dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\dfrac{4}{5}+\dfrac{5}{6}+\dfrac{6}{7}+\dfrac{7}{8}+\dfrac{8}{9}+\dfrac{9}{10}+\dfrac{10}{11}\ &+\dfrac{11}{12}+\dfrac{12}{13}+\dfrac{13}{14}+\dfrac{14}{15}+\dfrac{15}{16}+\dfrac{16}{17}+\dfrac{17}{18}+\dfrac{18}{19}+\dfrac{19}{20}+\dfrac{20}{21}\ &=\dfrac{1}{2}+\left(\dfrac{2}{3}-\dfrac{1}{3}\right)+\left(\dfrac{3}{4}-\dfrac{2}{4}\right)+\left(\dfrac{4}{5}-\dfrac{3}{5}\right)+\left(\dfrac{5}{6}-\dfrac{4}{6}\right)\ &+\left(\dfrac{6}{7}-\dfrac{5}{7}\right)+\left(\dfrac{7}{8}-\dfrac{6}{8}\right)+\left(\dfrac{8}{9}-\dfrac{7}{9}\right)+\left(\dfrac{9}{10}-\dfrac{8}{10}\right)+\left(\dfrac{10}{11}-\dfrac{9}{11}\right)\ &+\left(\dfrac{11}{12}-\dfrac{10}{12}\right)+\left(\dfrac{12}{13}-\dfrac{11}{13}\right)+\left(\dfrac{13}{14}-\dfrac{12}{14}\right)+\left(\dfrac{14}{15}-\dfrac{13}{15}\right)+\left(\dfrac{15}{16}-\dfrac{14}{16}\right)\ &+\left(\dfrac{16}{17}-\dfrac{15}{17}\right)+\left(\dfrac{17}{18}-\dfrac{16}{18}\right)+\left(\dfrac{18}{19}-\dfrac{17}{19}\right)+\left(\dfrac{19}{20}-\dfrac{18}{20}\right)+\left(\dfrac{20}{21}-\dfrac{19}{21}\right)\ &=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}+\dfrac{1}{11}\ &+\dfrac{1}{12}+\dfrac{1}{13}+\dfrac{1}{14}+\dfrac{1}{15}+\dfrac{1}{16}+\dfrac{1}{17}+\dfrac{1}{18}+\dfrac{1}{19}+\dfrac{1}{20}+\dfrac{1}{21}\ &=\sum_{n=2}^{21}\dfrac{1}{n}\ &=\ln 21 - \ln 1\ &=\ln 21 \approx 3.04452 \end{aligned}$$

三、结论

因此，这个分数序列的前 $20$ 项的和为 $\ln 21 \approx 3.04452$。我们可以发现，这个和并不是一个特别复杂的数，但是它的值可以用自然对数来表示。这也反映了数学上许多看似不相关的概念之间的深刻联系。