Can someone help me with this extra credit homework?

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Assigned: December 02, 2014 Due: December 04, 2014

Math 122 Extra Credit 13

In class, you saw that given a “nice” function f(x, y) and a “nice” region R in the xy-plane, you can switch the order of integration when you calculate∫∫

R

f(x, y) dA

Today we will look at a different example. Since integration is a generalization of addition, our example will simply be about addition.

1. Consider the real numbers aij for i = 1, 2, 3, . . . , j = 1, 2, 3, . . . given by

aij =

 

1 if i = j

−1 if i = j + 1 0 otherwise

The collection of the numbers aij can be visualized as a matrix 

1 0 0 0 0 · · · −1 1 0 0 0 0 −1 1 0 0 0 0 −1 1 0 0 0 0 −1 1 ...

. . . . . .

 

(a) Fix j and calculate ∞∑ i=1

aij . Then calculate ∞∑ j=1

( ∞∑ i=1

aij

) .

(b) Calculate ∞∑ j=1

a1j . Fix i > 2 and calculate ∞∑ j=1

aij . Then calculate ∞∑ i=1

( ∞∑ j=1

aij

) .

(c) Consider the statement

“If bij are real numbers for i = 1, 2, 3, . . . , j = 1, 2, 3, . . . , then

∞∑ j=1

∞∑ i=1

bij = ∞∑ i=1

∞∑ j=1

bij ”

Is the statement true or false (that is, can you always change the order of summation)? Use your results from parts (a) and (b) in your explanation.

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