## More Problems on Series

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

.

1.
Show that if b < 1, then the series is divergent.
2.
Show that if b > 1, then the series is convergent.
3.
What happens to if b=1?

Problem 7: Abel's Theorem
Let and be two sequences of real numbers such that

1.
there exists M such that for every , we have

;

2.
;

3.
the series is convergent.

Then the series is convergent.

Problem 8: Discuss the convergence or divergence of

.

Problem 9: Let be a sequence of positive decreasing numbers.

1.
Show that the sequence converges to 0, if the series converges. What about the converse?;
2.
Set

Is there a relationship between convergence of and ?;

3.
Assume that is convergent. What can you say about ?

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

.

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