Problems on Sequences
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 .