## 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 . [Trigonometry] [Calculus]
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996