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: Discuss the convergence or divergence of
,
where a and b are two parameters.
Problem 2: Discuss the convergence or divergence of
.
Problem 3: Discuss the convergence or divergence of
,
where
.
Problem 4: Discuss the convergence or divergence of
.
Problem 5: Discuss the convergence or divergence of
,
where a > 0.
Problem 6: Duhamel's Rule
Assume that the series satisfies
,
where b is a real number and the function satisfies
.
Problem 7: Abel's Theorem
Let and
be two sequences of real numbers
such that
;
Problem 8: Discuss the convergence or divergence of
.
Problem 9: Let be a sequence of positive decreasing
numbers.
Is there a relationship between convergence of and
?;
Problem 10: Let be a divergent series of positive
numbers. Discuss the convergence or divergence of the following
series:
where .
Problem 11: Discuss the convergence or divergence of
.
Problem 12: Discuss the convergence or divergence of
.
Problem 13: Discuss the convergence or divergence of
,
where a is a real number.
Problem 14: Discuss the convergence or divergence of
.