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I have difficulties in writing some equations in summation/product notation.

I want to write this in summation notation. $$p_1p_2+p_1p_3+...+p_1p_n+p_2p_3+p_2p_4+...+p_2p_n+...+p_{n-1}p_n$$

is it $$\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_ip_j \quad ?$$

or

$$\sum_{i=1}^{n-1}\sum_{j=1}^{n}p_ip_{i+j} \quad ?$$

or both wrong?

I want to write this in product notation. $$(1-p_1p_2)(1-p_1p_3)...(1-p_1p_n)(1-p_2p_3)(1-p_2p_4)...(1-p_2p_n)...(1-p_{n-1}p_n)$$

is it

$$\prod_{i=1}^{n-1}\prod_{j=i+1}^{n}(1-p_ip_j) \quad ?$$

or

$$\prod_{i=1}^{n-1}\prod_{j=1}^{n}(1-p_ip_{i+j}) \quad ?$$

or both wrong?

fairytale
  • 151

2 Answers2

2

For the sums: your first expression is correct, while your second expression should take the limit of the $j$-sum to be $n-i$.

Likewise the products: the limit of your second expression's $j$-product should be $n-i$.

As Exodd says, it's clearer just to use $$\sum_{1 \leq i < j \leq n}$$

Patrick Stevens
  • 36,135
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The first double sum $$\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_ip_j$$ is quite ok.

It is useful to recall the validity of \begin{align*} \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_ip_j=\sum_{1\leq i<j\leq n}p_ip_j=\sum_{j=2}^n\sum_{i=1}^{j-1}p_ip_j \end{align*}

The second double sum $$\sum_{i=1}^{n-1}\sum_{j=1}^{n}p_ip_{i+j}$$ is not correct. When looking at the term with index $i=n-1$ and $j=n$ we obtain $$p_{n-1}p_{2n-1}$$ which is not part of $$p_1p_2+p_1p_3+...+p_1p_n+p_2p_3+p_2p_4+...+p_2p_n+...+p_{n-1}p_n$$

We have quite the same situation when looking at the products. Again, it is useful to recall the validity of \begin{align*} \prod_{i=1}^{n-1}\prod_{j=i+1}^{n}(1-p_ip_j)=\prod_{1\leq i<j\leq n}(1-p_ip_j)=\prod_{j=2}^n\prod_{i=1}^{j-1}(1-p_ip_j) \end{align*}

Markus Scheuer
  • 108,315