5. (a) Let {an} be a sequence of integers with 0 < an < 9. Prove that > an10-" exists (and lies between 0 and 1). (This, of course, is the number which we usually denote by 0.a1azaza« . . . .)
5. (a) Let {an} be a sequence of integers with 0 < an < 9. Prove that > an10-" exists (and lies between 0 and 1). (This, of course, is the number which we usually denote by 0.a1azaza« . . . .)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve
![5. (a) Let {an} be a sequence of integers with 0 < an < 9. Prove that
an10- exists (and lies between 0 and 1). (This, of course, is the
number which we usually denote by 0.a,azaza4 . . . .)
(b) Suppose that 0 < x < 1. Prove that there is a sequence of integers
{an} with 0 < an < 9 and ) an10-" = x. Hint: For example,
a1 = [10x] (where [y] denotes the greatest integer which is < y).
(c) Show that if {an} is repeating, i.e., is of the form a1, az,
.. ak
a1, a2, · .. , ak, a1, a2, . . . , then
an10¬" is a rational number
(and find it). The same result naturally holds if {an} is eventually
repeating, i.e., if the sequence {an+*} is repeating for some N.
(d) Prove that if x = ) an10-n is rational, then {an} is eventually
repeating. (Just look at the process of finding the decimal expansion
of p/q-dividing q into p by long division.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76fcef72-833a-4962-928a-bced1874b6af%2Fafb4d590-62ce-42c5-a709-0431be29cfd4%2Fknpnkpd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. (a) Let {an} be a sequence of integers with 0 < an < 9. Prove that
an10- exists (and lies between 0 and 1). (This, of course, is the
number which we usually denote by 0.a,azaza4 . . . .)
(b) Suppose that 0 < x < 1. Prove that there is a sequence of integers
{an} with 0 < an < 9 and ) an10-" = x. Hint: For example,
a1 = [10x] (where [y] denotes the greatest integer which is < y).
(c) Show that if {an} is repeating, i.e., is of the form a1, az,
.. ak
a1, a2, · .. , ak, a1, a2, . . . , then
an10¬" is a rational number
(and find it). The same result naturally holds if {an} is eventually
repeating, i.e., if the sequence {an+*} is repeating for some N.
(d) Prove that if x = ) an10-n is rational, then {an} is eventually
repeating. (Just look at the process of finding the decimal expansion
of p/q-dividing q into p by long division.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
