Integral Test on Series

 

Motivation:

 

                       Some series are of the form as Riemann sum. So, it can be using integral to check the existence of the limit.  If the result of the integral goes to infinity,  then the Riemann sum itself does not converge.

 

                      For example,   S_n =  ,  will S_n converge when n  ?

                      Actually, the limit of S_n is the following integral

                                 

                                                   =

 

                      But if you tackle the problem directly, it is a little difficult:

 

                                                    S_n =

                      It is necessary to figure out    at first to find out the limit of S_n .  So, this opens a door to give some hints about using integral to find out the limits of some series.

 

 

                       Furthermore, sometimes, each term of a series can be associated to the value of a function at a point – it might be larger or less than the value of the function at the corresponding point. With the relationship, we might use the function to check the behavior of the series.

 

                       In order to handle those issues,  the first thing we will do is to have a broader definition on the interval for integral.  Previously, we only focus on the integral on a bounded interval [a,b] .  We will start with the following definition.

 

 

 

 

Definition:

              For the notation  ,  it is defined as

                                       = 

              Similarly, 

                                        = 

                                        =

 

 

               At the first glance,  the definition  is very simple and nature.   But when you consider to represent the integral as the limit of Riemann sum,  it turns out the “limit” of the result of another “limit” process.  So, it could be very powerful to handle a lot of situations.

 

 

 

 

 

Theorem ( Integral Test )

       Let  f(x) be a monotonically decreasing integrable function and f(x) > 0 for x  0. 

       If  a_n = f(n)  for n  N ,  then

 

                             converges    converges .

 

         Proof:

                        

                               Let  S_k =  .  Let’s consider the following diagram:

 

                                                 

                             Since f(x) is a monotonic decreasing function,  we can know from the area relationship above to get the following result:

 

                                                  = S_k

 

                                                

 

                             Furthermore,          from the diagram above.  Thus, we have

                                               

 

                          If    exists,   it means  the series {S_n} has upper bound. Furthermore, S_n is monotonic increasing ,  that implies  the limit of  S_n  exists.

 

                          On the other hand,  if   goes to infinity,   also goes to infinity.  Hence,  S_n diverges. With the result,  the convergence of  S_n can be fully determined by the integral .

 

 

 

Theorem ( p-series )

                    

                       S_k =  .    {S_k} converges only when p > 1 .

 

         Proof:

                            = ln(k)    when k  .

                        So, when p=1,  the series diverges.

                        

                        When p> 1, 

                                     =  =

                                   

                                     Thus, the series converges.

 

                        Hence {S_k} only converges when p > 1.

 

      

 

 

Summary for Convergence on Series

                          Up to this moment,  we just summarize the methods that can be used to determine if a series in R converges or not :  the Cauchy criterion can be used to check its tail properties;  if the series is monotonically increasing,  it is only necessary to check if it has a upper bound.  We can use a series that is known as convergent or divergent to compare with the series to check if it converges.  Ratio test and Root test can convert the problem into the question to determine the limit of a simple term – in this stage, L’Hospital rule probably will be used for finding the value of the limit.  Furthermore, the limit of Riemann sum are defined as integral – for the integral, we develop many method to find its value – that could also help to solve some problems on series.

 

                  A lot of series is just the value of Taylor expansion of a function at a given point – this kind of problems also can be quickly solved if the function can be quickly identified.  Along with the integral test introduced here,  a lot of interesting series can be generated from those methods.

 

                  There are other interesting topics on the study related to series: Fourier analysis, which is widely used on Engineering, eigenvalue problems, solutions to ordinary differential equations or partial differential equations, complex analysis… Those who are interested in those topics can find many literatures associated with them.