(a) Show that for all m ≥ N |(r1 + r2 + · · · + rm) − S| ≤ epsilon. Conclude that the iterated sum sigma∞i=1 sigma∞j=1 aij converges to S. (b) Finish the proof by showing that the other iterated sum,sigma∞i=1 sigma∞j=1 aij converges to S as well.Notice that the same argument can be used once it is established that,for each fixed column j,the sum sigma∞i=1 converges to some real number cj . One final common way of computing a double summation is to sum alongdiagonals where i + j equals a constant. Given a doubly indexed array {aij :i, j ∈ N}, let d2 = a11, d3 = a12 + a21,d4 = a13 + a22 + a31,and in general set dk = a1,k−1 + a2,k−2 + · · · + ak−1,1. Then, sigma∞k=2 dk represents another reasonable way of summing over every aij in the array.
(a) Show that for all m ≥ N
|(r1 + r2 + · · · + rm) − S| ≤ epsilon.
Conclude that the iterated sum sigma∞i=1 sigma∞j=1 aij converges to S.
(b) Finish the proof by showing that the other iterated sum,sigma∞i=1 sigma∞j=1 aij converges to S as well.Notice that the same argument can be used once it is established that,for each fixed column j,the sum sigma∞i=1 converges to some real number cj .
One final common way of computing a double summation is to sum along
diagonals where i + j equals a constant. Given a doubly indexed array {aij :
i, j ∈ N}, let d2 = a11, d3 = a12 + a21,d4 = a13 + a22 + a31,and in general set
dk = a1,k−1 + a2,k−2 + · · · + ak−1,1.
Then, sigma∞k=2 dk represents another reasonable way of summing over every aij in the array.
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