Problems on Sequences

Posted By on December 28, 2014


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Some Special Limits
Introduction to Series

In this page you will find some not so easy problems on sequences.

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 thatdisplaymath97.

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 .

Some Special Limits
Introduction to Series

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Posted by Akash Kurup

Founder and C.E.O, World4Engineers Educationist and Entrepreneur by passion. Orator and blogger by hobby

Website: http://world4engineers.com