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Sn=∑i=1naiSn′=∑i=1n1aiS(n)=∑i=1n(Sn′ai−1) \begin{aligned} S_{n}&=&\displaystyle\sum_{i=1}^{n} a_i \\ S_{n}^{'}&=&\displaystyle \sum_{i=1}^{n} \dfrac{1}{a_i} \\ S(n)&=&\displaystyle\sum_{i=1}^{n} (S_{n}^{'}a_i - 1) \end{aligned} SnSn′S(n)===i=1∑naii=1∑nai1i=1∑n(Sn′ai−1)
I define three kinds of very similar functions as above.
Find the value of S(1000)−S1000S1000′S(1000)-S_{1000}S_{1000}^{'}S(1000)−S1000S1000′.
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