Calculate a Product Sum
Calculator for product sums, a sum Σ of products Π. The inner product with the running variable j runs from o to p, the outer sum with the running variable i runs from m to n. Only the variables i and j may occur in the sum term. The basic arithmetic operations + - * / are allowed, as well as the power function, like pow(j#i) for ji. Further allowed functions are sin(), cos(), tan(), asin(), acos(), atan(), log() for the natural logarithm and fac() for the factorial. Also, the constants e and pi can be used. If i≠j is chosen, factors with equal running variables are omitted. The double product can quickly lead to very high values.
Using the preset values, a simple product sum is calculated. At each step, the sum of the two counter variables i and j is calculated. These sums are 2, 3, 4, 5, and 6 in the first row. These initial five individual sums, corresponding to the number of steps for j, are multiplied by each other step by step. This yields the first intermediate result, which in this case is 720. The process then continues with the next five individual sums, 3, 4, 5, 6, and 7, which are then multiplied again to obtain the next intermediate result, here 2520. The two intermediate results are added together to obtain 3240. This is done only once again, because the number of steps for i is three. The third intermediate result is 6720, and the sum of all intermediate results, and thus the final result in this example calculation, is 9960.
Besides the product sum, there is also the sum product, where the calculation is reversed. There, the inner sums are multiplied by each other. Both calculators use the same example values, allowing for a good comparison of the behavior of the product sum and the sum product. It is impossible to say definitively which method leads to faster increasing or converging results. This depends on the values used and the number of individual additions and multiplications. For the given example values, the product sum is higher up to the second row, after which the sum product surpasses it.