More Problems on Sequences

In this page you will find some not so easy problems on sequences. We invite you to solve them and submit the answer to SOS MATHematics. We will publish your answer with your name. Good luck.

Problem 1: Let tex2html_wrap_inline87 be a sequence of real numbers such that

displaymath89.

Show that

displaymath91.

A generalization of this result goes as follows:


Let tex2html_wrap_inline87 and tex2html_wrap_inline95 be sequences of real numbers such that

displaymath97.

Then, we have

displaymath99.

Problem 2: Evaluate

displaymath101.

Problem 3: Discuss the convergence of

displaymath103.

Problem 4: Discuss the convergence of

displaymath105.

Problem 5: Let tex2html_wrap_inline107 and tex2html_wrap_inline109 be two sequences of integers. Assume tex2html_wrap_inline111, for all tex2html_wrap_inline113 , and tex2html_wrap_inline115 converges to an irrational number. Show that

displaymath117.

Problem 6: Let tex2html_wrap_inline119 (that is tex2html_wrap_inline121 ). Show that there exists a unique real number x such that

displaymath125.

Call this number tex2html_wrap_inline127 . Show that

displaymath129.

Problem 7: Evaluate

displaymath131.

Use it to show that

displaymath133.

Problem 8: Let tex2html_wrap_inline135 be a real number such that tex2html_wrap_inline137 . Set

displaymath139.

Find the limit of tex2html_wrap_inline87 .

Problem 9: Let tex2html_wrap_inline87 be a sequence of real numbers such that

displaymath145

whenever tex2html_wrap_inline147 . Assume that tex2html_wrap_inline149 . Show that

displaymath151

is convergent.

Problem 10: Let tex2html_wrap_inline87 be a sequence of real numbers such that

displaymath155

Show that the sequence

displaymath157

either converges to its lower bound or diverges to tex2html_wrap_inline159 .

[Trigonometry] [Calculus]
[Geometry] [Algebra] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

Copyright 1999-2014 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour