## 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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*Mohamed A. Khamsi *

Tue Dec 3 17:39:00 MST 1996

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