# Infinite Summation Portfolio. I will consider the general sequence with constant values

IB Mathematics SL Type 1: Infinite Summation Portfolio

The English School

IB Mathematics SL

Math Portfolio (Type1)

Infinite Summation

Candidate Code:000185

19th of October 2011

Bogotá, Colombia

Aim: The aim of this portfolio is to research about the summation of infinite series. In first place I will consider the general sequence with constant values for  and variables for . The , which is calculate the sum  of the first  terms of the above sequence for  will be calculated with the programme of Microsoft Excel and illustrated in graphs with Graphmatica; another computer based program, so that we can create a general statement which can be proven. Furthermore, to expand the investigation I will explore the same general sequence, but with variables in both  and . I will accomplish this by doing the same method as the first exercises. Finally, I will conclude by showing the scopes/limitations of the general statement.

Where,

, ,  ,   ,

We must take into account that factorial notation is in succession, this means that the factorial notation shows all the natural numbers from 1 to , in such way that:

Following the first exercise, we will change the variables of terms and , in such way that and . If we replace these values into the original sequence we obtain:

, ,  ,  , …

In order to calculate the sum  of the first  terms of the above sequence for  we must know that the sum of the first 10 values is in progression. Hence, I will use Microsoft Excel in order to plot results in a suitable table. The first column will contain the different values of , which come from 1 to 10. The second column will contain the results obtained by replacing each of the values in the form of . And the third column will contain the gradual sum of each of the terms obtained in the second column. For example, the first value of the third column will be added to the second value of the second column giving the second value of the third column, and so on.

With the help of Excel the table will look like this:

Table 1: Shows the sum of the first 10 terms for

After analysing the previous results the sum  of the first  terms of the above sequence for  is 2.

Likewise, we can represent the relationship between  and  in a graph by replacing the values of  in the y-axis and the values of  in the x-axis can do this. The graph should look like this:

Table 2: Shows the values of the graph of  against  of

After using the computer programme Graphmatica, the result of the graph is this:

Graph 1: Shows the relation between  and  in

We can observe that the  values increase exponentially until they reach the number 2. At this point they stabilize. Hence, we can conclude that this sequence is convergent. In other words, it is approaching a definite limit as more of its terms are added. This happens because the  values are getting smaller rather than being bigger due to the factorial notation in the denominator. As the denominator gets bigger we obtain results with smaller numbers. From this information we can say that the  values tend to move to 2 if the variable is , likewise the way  approaches to .

Next, we can consider the same sequence with , but changing the variable  to . The result of the sequence is as following:

, ,  ,  , …

Now that we know the sequence we can now get the sum  of the first  terms of the above sequence for . Like the sequence before, I will use Microsoft Excel in order to get the table and therefore get the sum of all terms.

The table is as following:

Table 3: Shows the sum of the first 10 terms for

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