1549901369-Elements_of_Real_Analysis__Denlinger_

8.8 *Elementary Transcendental Functions (Project) 53 5

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17. For the double series I: aij defined by the matrix
    i,j=l

0 -1 0 0 0

1 0 -1 0 0

0 1 0 -1 0

0 0 1 0 -1...

0 0 0 1 0

find the sum by rows and the sum by columns.

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* 18. Write the infinite matrix representing the "double geometric series" I: ri sj.

*19.

i,j=O

Assuming 0 < Iii, IJI < 1, find the sum by rows (or columns).

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We could say that a double series I: aij converges to S ¢:;> Ve > 0, 3n 0 E

i,j=l

N 3 m,n ::'.:: no ==;. li~j~l aij -SI < E:. In this definition, i~j~l aij

plays the role of "partial sum." Explain why this method of summing a

double series could be called "summing by upper left rectangles." Prove

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that if the sum by rows (or the sum by columns) of I: laijl converges,

i,j=l

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then I: aij converges by this definition. Show that the double series of

i,j=l

Exercise 17 does not converge by this definition.

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*20. Given series I: ak and I: bk, find an infinite matrix whose sum by

8.8

k=l k=l

columns is their Cauchy product.

* Element ary Transcendental Functions

(Project)

This section is designed as a project to be completed by stu-

dents. It has no exercise set at the end. Instead, students are

asked to furnish proofs for claims left unproved in the text.