Problems on Sequences

Posted By on December 28, 2014

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

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

.

Show that

.

A generalization of this result goes as follows:

Let and be sequences of real numbers such that.

Then, we have

.

Problem 2: Evaluate

.

Problem 3: Discuss the convergence of

.

Problem 4: Discuss the convergence of

.

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

.

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

.

Call this number . Show that

.

Problem 7: Evaluate

.

Use it to show that

.

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

.

Find the limit of .

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

whenever . Assume that . Show that

is convergent.

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

Show that the sequence

either converges to its lower bound or diverges to .

Posted by Akash Kurup

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

Website: http://world4engineers.com