Problems " in determining the convergence of infinite series

In this article we are presented different approaches to determine the convergence of infinite series. Students often do not know which criterion is appropriate for determining the convergence of the series. We don ́t know always to determine the convergence of the series with using some criterions and so it is sometimes appropriate to use more of the given criterions.


INTRODUCTION
Within the subject Mathematics 2 the students of the Faculty of Engineering Slovak University of Agriculture in Nitra deal with the study of infinite numerical and functional series.The first we define why infinite series is convergent and then proceed with to determining the convergence of series.Students gain knowledge that serve to determine of convergence criteria.Students in solving of the task within the exercise or the test have often problem and they don´t know which criteria can be used.In some examples, that convergence series can be determined by various criteria.However, students often find that the criterion that they have chosen does not lead to the identification of convergence.

MATERIAL AND METHODS
Now we list the definition of infinite numerical series and the n-th partial sum.Definition 1.When we wish to find the sum of an infinity sequence    1 n n a , we call it an infinity series and write it in form The often is difficult when student with its knowledge does not know always determine n-th partial sum of the series and use it to check the convergence series.It is therefore necessary to determine the convergence another way.For its determination we use the convergence criteria.Now we list necessary condition of convergence series.
This theorem is necessary condition of convergence series and the opposite theorem not valid, so if then the series does not converges.
We will deal only series with non-negative members, so , 0 and criteria for the determination of convergence.We most often use the following convergence criteria: 1. Comparison test 2. Cauchy limit criterion 3. D´Alembert limit criterion 4. Integral test Now we give their definitions.

Comparison test:
Let

Cauchy limit criterion:
Let the series   1 n n a is the series with nonnegative terms and there´s , then the series diverges.

D´Alembert limit criterion:
Let the series   1 n n a is the series with positive terms and there´s q a , then the series diverges.

Integral test:
Let the series   1 n n a is the series with nonnegative terms and let valid ... ...
A is number, then the series converges and for his sum s valid

RESULTS AND DISCUSSION
Now we give the task., therefore , thus series could be convergent.We have to find out with other criteria.As the n-th member is not n-th power, so student does not using Cauchy limit criterion.Probably the student not use comparison test because he does not know which the series should be compared.Students are "most like" D'Alembert limit criterion for its relative simplicity.
The n-th member of the series n n According to this criterion, we cannot decide whether it the series We must therefore determine convergence otherwise.The n-th member of the series modify as follows For the n-th partial sum of the series n s is valid To determine the convergence of the series could have been used also integral test, because for the members of the series apply: ... is continuous and decreasing (hence non-increasing) on the interval ) , 1  (see Figure 1).

CONCLUSIONS
In the article we showed different possibilities of determining the convergence of infinite numerical series.Students not only of the Faculty of Engineering Slovak University of a) if the infinite series   1 n n b converges, then the infinite series   1 n n a also converges, b) if the infinite series   1 n n b diverges, then the infinite series   1 n n a also diverges.When using this test, we comparative the series whit series 

Task.
Check the convergence of series 

Fig. 1
Fig. 1 Graph of the function