Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
If g (0, ∞) → (0, ∞) is continuous with limx→0 g(x) = +∞o and Σ an is converging series with strictly positive terms, then Σ(1/g(an)) is a converging series. Select one: O a. False, here is a counter-example: an = 1/n², g(x) = 1/(nx). O b. True, because limx→o g(x) = +∞o implies g(x) ≥ M/x for some constant number M > 0, so 1/g(an) ≤an/M and, by comparison theorem, 1/g(an) converges. O c. True, because by n-th term test an → 0 and thus, by sequential characterisation of the continuity, 1/g(an) 0, so 1/g(an) must converge. O d. False, here is a counter-example: an = 1/n², g(x)= = x-1/3

If g (0, ∞) → (0, ∞) is continuous with limx→0 g(x) = +∞o and Σ an is converging series with strictly positive terms, then Σ(1/g(an)) is a converging series. Select one: O a. False, here is a counter-example: an = 1/n², g(x) = 1/(nx). O b. True, because limx→o g(x) = +∞o implies g(x) ≥ M/x for some constant number M > 0, so 1/g(an) ≤an/M and, by comparison theorem, 1/g(an) converges. O c. True, because by n-th term test an → 0 and thus, by sequential characterisation of the continuity, 1/g(an) 0, so 1/g(an) must converge. O d. False, here is a counter-example: an = 1/n², g(x)= = x-1/3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Need help with this question. Thank you :)

 

If g : (0, ∞) → (0, ∞) is continuous with limx→0 g(x) = +∞o and Σan is converging series with strictly positive terms, then Σ(1/g(an)) is a converging series. Select one: a. False, here is a counter-example: an = 1/n², g(x) = 1/(nx). b. True, because limx→o g(x) = +∞ implies g(x) ≥ M/x for some constant number M > 0, so 1/g(an) ≤ an/M and, by comparison theorem, Σ 1/g(an) converges. c. True, because by n-th term test an → 0 and thus, by sequential characterisation of the continuity, 1/g(an) → 0, so Σ 1/g(an) must converge. O d. False, here is a counter-example: an = x² = x-1/3. 1/n², g(x) =
Transcribed Image Text:If g : (0, ∞) → (0, ∞) is continuous with limx→0 g(x) = +∞o and Σan is converging series with strictly positive terms, then Σ(1/g(an)) is a converging series. Select one: a. False, here is a counter-example: an = 1/n², g(x) = 1/(nx). b. True, because limx→o g(x) = +∞ implies g(x) ≥ M/x for some constant number M > 0, so 1/g(an) ≤ an/M and, by comparison theorem, Σ 1/g(an) converges. c. True, because by n-th term test an → 0 and thus, by sequential characterisation of the continuity, 1/g(an) → 0, so Σ 1/g(an) must converge. O d. False, here is a counter-example: an = x² = x-1/3. 1/n², g(x) =
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS