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You can interchange the order of a double sum when the double summation converges absolutely. This means that the sum of the absolute values of the terms in the series is finite. If a doubly-indexed infinite sequence is absolutely summable in some way, you may change the order of summation and obtain the same result.

In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or ...

Suppose that I want to sum all elements in the matrix. Then I could sum all rows per column, or all columns per row, i.e. xij, as the order of summation does not matter.

Considering the sum ??i=0??j=0aij, in general we are not allowed to interchange the summation order (i.e. pass to ??j=0??i=0aij) but some more hypothesis are required to allow the change of order (e.g., by Fubini Theorem, we are allowed if ??i=0??j=0|aij|<?).

How to Shift the Index of Summation with Infinite Series - YouTube YouTube Start of suggested clip End of suggested clip Okay. So now we're gonna go ahead and replace this sum with whatever we end up with. So can you callMoreOkay. So now we're gonna go ahead and replace this sum with whatever we end up with. So can you call it K solve friend call it k solve for n good stuff let's rewrite our sum. So we have the sum. Now.

Considering the sum ??i=0??j=0aij, in general we are not allowed to interchange the summation order (i.e. pass to ??j=0??i=0aij) but some more hypothesis are required to allow the change of order (e.g., by Fubini Theorem, we are allowed if ??i=0??j=0|aij|<?).

Answer and Explanation: We can switch the double sums when we need to swap the order of summation. Switching double sums does not affect our results. Let there be a double summation ? i = 1 ? ? j = 1 ? F ( i , j ) , its equivalent swapped summation will be ? ? j = 1 ? ? i = 1 j F ( i , j ) .

Suppose that I want to sum all elements in the matrix. Then I could sum all rows per column, or all columns per row, i.e. xij, as the order of summation does not matter.