My question is simple: How one can simplify this product of sums:

$$S=\left(\sum_{k=1}^{n}a_{k}\right) \left(\sum_{k=1}^{p}b_{k}\right) \left(\sum_{k=1}^{q}c_{k}\right)$$

where $a_{k},b_{k},c_{k}$ are real sequences and $n<p<q$.

  • 1
    $\begingroup$ What do you mean by simplify? I think that is as compact as things can get without knowing a,b,c. If you want to expand them you can assign different indices, but I wouldn't call that simpler! $\endgroup$ – Kitter Catter Nov 3 '16 at 17:51
  • $\begingroup$ Can you find an alternative expression simplier than $a(b_1+b_2)(c_1+c_2+c_3)$? $\endgroup$ – Ng Chung Tak Nov 3 '16 at 17:53

Don't you have $$ S=\left(\sum_{k=1}^{n}a_{k}\right) \left(\sum_{k=1}^{p}b_{k}\right) \left(\sum_{k=1}^{q}c_{k}\right) = \sum_{i=1}^n \sum_{j=1}^p \sum_{k=1}^q a_i b_j c_k? $$


Is this what you want? \begin{align*} (\sum_{k=1}^{n}a_{k})(\sum_{k=1}^{p}b_{k})(\sum_{k=1}^{q}c_{k}) = \sum_{i=1}^n\sum_{j=1}^p\sum_{k=1}^q a_ib_jc_k. \end{align*}

  • 5
    $\begingroup$ This is about all you can do with such general sequences. It's frankly debatable which is "simpler". $\endgroup$ – Gabriel Burns Nov 3 '16 at 17:52

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