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 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 .

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